I enjoyed the music in Tribes 2 so I let that play, but I'd play Counter-Strike in silence so I could hear footsteps, and I think that's carried over to other games. Nowadays I mostly play Heroes of the Storm and I don't think I've ever played with music -- guess I think it'd be distracting even if I'm not trying to listen for 3D audio cues.
While working/driving, I usually listen to mashups like
or electro mixes like
and if I listened to music while playing competitive games, I'd probably listen to stuff like these.