It it appropriate to request a letter of recommendation via LinkedIn? I had a professor who told me during the year that he'd be happy to write me a letter of recommendation for a research degree I was contemplating applying to. I've since decided to apply.
My university (in Europe) is on break until 1 October. I emailed the professor asking for the letter three weeks ago on his university email address. He hasn't answered me. However, I sent him a connection request on LinkedIn, and he accepted it within 45 seconds. I am thus wondering if he is not checking his university email during our vacation. However, I would like to know if he is willing to write it before October, as waiting until then would give me insufficient time to request one from someone else.
Would it be inappropriate for me to send him a follow-up letter request via LinkedIn, since I know he's checking that? Would it be more appropriate to make reference to the email I sent him, or send my entire request via the messenger? Or should I limit myself to communication via the university email address and just hope he reads it?
If someone is on LinkedIn, especially if they are active on LinkedIn, I see no reason why they would be offended if you sent them a message on LinkedIn. LinkedIn is a professional networking site. That would be exactly what you are doing. I would write something straight forward:

[Salutation]
I am applying to graduate programs at a few schools. I enjoyed your class in Underwater Heavy Metal Guitar and felt I did well in the class. Would you be willing to write me a letter of recommendation for my graduate program applications?
Sincerely,
-bron10

Three weeks is a long enough time that this professor has likely not seen (or has forgotten about) your email. There is no reason to cower in fear about asking for a letter of recommendation from a professor who said he would write you one. If he is at all reasonable, a simple message on LinkedIn will be just fine.