Gotta agree with the praise for JazzMusique. I’m streaming either that or (more often) HouseMusique during nearly all waking hours.
I’m with you on the “not paying attention” to the audio, so I found a solution. There’s a piece of software called iNet Streamer for Mac OS X, which allows you to record all the streaming radio directly to your hard drive for further review. That way, the music stays ambient, and you can find cool new artists later.