For games like Tenchu, I usually go with something like Abingdon Boys School or maybe the occasional T.M.Revolution song.
For FPS games, I always find 3OH!3, Hadouken!, Cobra Starship or other such artists good. That's a seriously strange phenomenon if you ask me.
As for Sonic (because yes, I'm truly still a bit of a Sonic fangirl. It's been installed in me since birth, it's like a genetic disease) Crush 40 is really the only band to listen to.
What I do is I have the effects turned up on the games and yet the game music is off - I like to hear enemies but I like my own music too.