I recommend some kind of cycling-specific short. I have had good results wearing normal bike shorts under clothing, and believe that the underwear versions would work well too. One thing I wouldn't recommend would be something with straps, which is what the athletic supporter-type things I've had in the past had. I'd strongly prefer something that was smooth and even.

If you don't want something padded, there are compression shorts, I think designed to go under other sports wear, that would work quite nicely. These would keep the guys out of the way, be smooth with minimal seams or other things to create friction, and would wick away sweat and not become soggy. I've worn similar for running, though I prefer padded shorts for riding. http://www.amazon.com/Champion-Compr.../dp/B00AKSCK2I