What is the buddhist way of responding to compliments?
Just because you're Buddhist doesn't mean you can't say "thank you" :P

All facetiousness aside, Buddhism is clear about the practice.  For instance, it stresses compassion and non-attachment, so evaluate your course of action against those two.


If someone compliments you and you don't politely acknowledge the
compliment, this person may be offended.  That isn't compassionate of you.
If someone compliments you and you let it get to your head, you could
be fostering attachment towards self, the thing about which you were
complimented, and the praise of others.


So how can you approach compliments in a way that takes the above into account?  Well, you could thank the person while remembering that you are thanking them out of compassion and consideration for their feelings and not because they gave you some validation you needed.

Ultimately, it's about skillful means.  Don't seek every possible situation in Buddhist texts; rather, make sure you understand the principles and use that understanding to guide your actions.