My niece Joanna sent me a link to Pandora Radio last year, and I've been really loving it. You can create your own station based on a single artist that you like, or mix it all up.
I have a couple of stations that I listen to all the time, one based on Over the Rhine and one for Fernando Ortega. When I'm playing with Anna in the computer room we listen to one based on Elizabeth Mitchell, and when I want Spanish music I listen to one like Ricardo Arjona.
To make a new station, you just have to tell them who you like, and they'll analyze the music and find lots of things that are similar. You can thumbs up or down the individual songs to give further direction. I used to listen to Yahoo radio, but I like Pandora better. No commercials and no log in.
You can share stations too, so let me know if you want one of mine. Bear in mind, I'm kind of an old lady.