user760: What does Buddhism say about polyamory? By polyamory I mean a consensual romantic relationship that involves more than two adult people, which may or may not involve sexual contact.
Is it wholesome, unwholesome or neutral? Is it any different than a romantic relationship between two people? Do different branches or traditions have different views on this subject?

user-1: So far as I know, "Buddhism" has no views on sexual norms. The Pali Canon mentions different types of gender, but seems to be non-judgemental about different kinds of marriage. In traditional Tibet, for example, a woman might marry two brothers. Logically, all varieties of sexuality must ultimately be abandoned, but the Buddha had important married householder followers, some of whom even attained arhantship. 
One might assume that the Buddha would counsel married householders to respect each other and practice restraint, but this does not imply a preference for one sort of "arrangement" over another. The Pali Canon seems to suggest that wrongful sexuality or sexual misconduct (the third precept) implies adultery. "Sexual misconduct" and "wrong behaviours" are referred to, but not defined.

user-1: The five precepts, which include sexual misconduct, are often said to be open to interpretation. If an open relationship works for everyone concerned, then I'd say it's likely no different from a homosexual one. 
But, I don't think that as long as one does not deceive anyone then you're fine. Take e.g. this in Sexuality in Classical South Asian Buddhism, p521

I've not read the book, but would guess that simply being up-front would not be enough: that if one of the parties would prefer a monogamous relationship then it's a form of misconduct.
You are violating the wishes and so commitment of your partner if they don't have a reasonable say in who else you are sexually involved with. It's not really a complex issue, ethically, only inter-subjectively.

user8579: Ignoring cultural beliefs and traditions and thinking in terms of pure Buddhism, it can be a sin and at the same time can not be a sin.
Polyamory is not a sin if both your partners knows what is exactly happening and both of them know that you are in a polyamory. Hence if either of the partners do not like it, they have a choice to leave. You have been honest.
If you mention to both of your partners that you are monogamous yet you continue the polygamous relationship, then you are deceiving both of your partners ( because each of them thinks they are the only one you have). This is a sin. Assuming one day both of them find out that you did, then you have caused pain and anger to both of your partners, that is also a sin. 
Take a look at the 5 precepts: 

harming living things.
taking what is not given.
sexual misconduct.
lying or gossip.
taking intoxicating substances eg drugs or drink.

The third precept says to refrain from sexual misconduct. So assuming your partners are aware of the polygamous nature in your relationship but if one or all of them happened to be spouses of another individual (or happened to be in a relationship), then you are violating the 3rd of the 5 precepts as you have caused adultery.
You would immediately ask me why? Well it is because, although you wont hurt your partners, the other partners of your partners does not know it. They are deceived. Therefore that is a sin. It is true that your partners made that choice of getting into the polygamous relationship with you ignoring their spouses or boyfriend/girlfriend. In this scenario, your partners will commit a sin for deceiving their spouse or the person with the relationship, and you will face the karma of committing adultery because you knew they were married/engaged/in a relationship.

user7149: As far as the core of Buddhism is concerned, anything that causes craving is to be abandoned, if your ultimate goal is Enlightment or becoming an Arhat.
There maybe some lenient rule regarding this in different forms of Buddhism, but those are just for the lay followers, someone who just follows Buddhism without any stringent aim..
So it depends upon what you want to become and to what extent you want to follow Buddhism.
You can always find excuses like this in atleast any one form of Buddhism.
Making excuses just makes you a lay follower.
It doesn't lead to cessation of suffering or give knowledge.
If you think my answer is irrelevant, you are free to downvote :)

user-1: 
What does Buddhism say about polyamory?

Buddhism is not conserned with the title of the romantic relationship or the sexual orientation of the partners in it.
Buddhism goes much deeper than that, it looks instead at the defilements that are connected with romantic relationships, e.g. "lust, craving, desire, jealousy, hatred etc." 

Is it any different than a romantic relationship between two people?

It is not. The above mentioned defilements, are the same no matter what kind of romantic relationship we are talking about.

Is it wholesome, unwholesome or neutral?

According to the Abhidhamma, there are 52 mental factors (cetasikas). 14 of these are unwholesome mental factors, among those are "hatred, jealousy, avarice, craving". They are of course also part of the 3 root defilements, i.e. greed, hatred and delusion. 
If one or more of these mental factors are present, it becomes unwholesome.

Do different branches or traditions have different views on this subject?

Good question. I do not have an answer to that. I would think that it's the same opinion, since we are dealing with the root defilements. I might be wrong about this though.  

760: <p>What does Buddhism say about polyamory? By polyamory I mean a consensual romantic relationship that involves more than two adult people, which may or may not involve sexual contact.</p>

<p>Is it wholesome, unwholesome or neutral? Is it any different than a romantic relationship between two people? Do different branches or traditions have different views on this subject?</p>
 -1: <p>So far as I know, "Buddhism" has no views on sexual norms. The Pali Canon mentions different types of gender, but seems to be non-judgemental about different kinds of marriage. In traditional Tibet, for example, a woman might marry two brothers. Logically, all varieties of sexuality must ultimately be abandoned, but the Buddha had important married householder followers, some of whom even attained arhantship. </p>

<p>One might assume that the Buddha would counsel married householders to respect each other and practice restraint, but this does not imply a preference for one sort of "arrangement" over another. The Pali Canon seems to suggest that wrongful sexuality or sexual misconduct (the third precept) implies adultery. "Sexual misconduct" and "wrong behaviours" are referred to, but not defined.</p>
 -1: <p>The five precepts, which include sexual misconduct, are often said to be open to interpretation. If an open relationship works for everyone concerned, then I'd say it's likely no different from a homosexual one. </p>

<p>But, I don't think that as long as one does not deceive anyone then you're fine. Take e.g. this in Sexuality in Classical South Asian Buddhism, p521</p>

<p><a href="https://i.stack.imgur.com/7ZLtj.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/7ZLtj.png" alt="enter image description here"></a></p>

<p>I've not read the book, but would guess that simply being up-front would not be enough: that if one of the parties would prefer a monogamous relationship then it's a form of misconduct.</p>

<p>You are violating the wishes and so commitment of your partner if they don't have a reasonable say in who else you are sexually involved with. It's not really a complex issue, ethically, only inter-subjectively.</p>
 8579: <p>Ignoring cultural beliefs and traditions and thinking in terms of pure Buddhism, it can be a sin and at the same time can not be a sin.</p>

<p>Polyamory is not a sin if both your partners knows what is exactly happening and both of them know that you are in a polyamory. Hence if either of the partners do not like it, they have a choice to leave. You have been honest.</p>

<p>If you mention to both of your partners that you are monogamous yet you continue the polygamous relationship, then you are deceiving both of your partners ( because each of them thinks they are the only one you have). This is a sin. Assuming one day both of them find out that you did, then you have caused pain and anger to both of your partners, that is also a sin. </p>

<p>Take a look at the 5 precepts: </p>

<ul>
<li>harming living things.</li>
<li>taking what is not given.</li>
<li><strong>sexual misconduct.</strong></li>
<li>lying or gossip.</li>
<li>taking intoxicating substances eg drugs or drink.</li>
</ul>

<p>The <strong>third precept</strong> says to refrain from sexual misconduct. So assuming your partners are aware of the polygamous nature in your relationship but if one or all of them happened to be spouses of another individual (or happened to be in a relationship), then you are violating the 3rd of the 5 precepts as you have caused adultery.</p>

<p>You would immediately ask me why? Well it is because, although you wont hurt your partners, the other partners of your partners does not know it. They are deceived. Therefore that is a sin. It is true that your partners made that choice of getting into the polygamous relationship with you ignoring their spouses or boyfriend/girlfriend. In this scenario, your partners will commit a sin for deceiving their spouse or the person with the relationship, and you will face the karma of committing adultery because you knew they were married/engaged/in a relationship.</p>
 7149: <p>As far as the core of Buddhism is concerned, anything that causes craving is to be abandoned, if your ultimate goal is Enlightment or becoming an Arhat.</p>

<p>There maybe some lenient rule regarding this in different forms of Buddhism, but those are just for the lay followers, someone who just follows Buddhism without any stringent aim..</p>

<p>So it depends upon what you want to become and to what extent you want to follow Buddhism.</p>

<p>You can always find excuses like this in atleast any one form of Buddhism.
Making excuses just makes you a lay follower.
It doesn't lead to cessation of suffering or give knowledge.</p>

<p>If you think my answer is irrelevant, you are free to downvote :)</p>
 -1: <blockquote>
  <p>What does Buddhism say about polyamory?</p>
</blockquote>

<p>Buddhism is not conserned with the title of the romantic relationship or the sexual orientation of the partners in it.</p>

<p>Buddhism goes much deeper than that, it looks instead at the defilements that are connected with romantic relationships, e.g. <em>"lust, craving, desire, jealousy, hatred etc."</em> </p>

<blockquote>
  <p>Is it any different than a romantic relationship between two people?</p>
</blockquote>

<p>It is not. The above mentioned defilements, are the same no matter what kind of romantic relationship we are talking about.</p>

<blockquote>
  <p>Is it wholesome, unwholesome or neutral?</p>
</blockquote>

<p>According to the <a href="http://www.accesstoinsight.org/lib/authors/mendis/wheel322.html#cetasika" rel="noreferrer">Abhidhamma</a>, there are 52 mental factors (cetasikas). 14 of these are unwholesome mental factors, among those are <em>"hatred, jealousy, avarice, craving"</em>. They are of course also part of the 3 root defilements, i.e. <em>greed, hatred and delusion</em>. </p>

<p>If one or more of these mental factors are present, it becomes unwholesome.</p>

<blockquote>
  <p>Do different branches or traditions have different views on this subject?</p>
</blockquote>

<p>Good question. I do not have an answer to that. I would think that it's the same opinion, since we are dealing with the root defilements. I might be wrong about this though.  </p>
