What to do if authors don't respond to my serious concerns about their paper? I recently read a paper that conducted an experiment, analyzed it, and reached a conclusion. However, the way they conducted the analysis is seriously flawed and cannot be used to support the conclusion.

As far as I can tell, the experiment is valid and only the analysis is problematic. Thus the paper can be rewritten, although the conclusion may completely change. 

The paper is published in a highly-reputable and prestigious scientific journal. The authors are all senior researchers at reputable institutions. 
It's a bit of a surprise that this flaw got past the authors and peer review. I suspect that, because the conclusion confirms what many people already believe, the analysis was not scrutinized too closely. I only became suspicious of it because the measured effect was too strong. The analysis is also reasonably complex and the flaw is somewhat subtle.

I contacted all three authors by email and explained the problem with their analysis. I did by best to phrase the email appropriately.

A month later, I have received no response to my email. What would be a reasonable course for further action? Options include:


Send the authors a follow-up email. (If so, what should I say to get the message across?)
Contact the journal with my concerns.
Write a response. (Would such a thing get published?)
Do nothing. (I think the paper is too important.)
Write a response. (Would such a thing get published?)
  


This. Such things are usually titled "Comment to..." and, yes, they are published, typically alongside with a reply from the authors of the commented paper (the comment is usually sent to them by the journal editor).

As usual disclaimer, since things may vary across fields and journals, check if the journal in question has already published comments of this type and, in doubt, contact the editor.