First there was Napster. When that fell, we had Kazaa and a host of imitators. Pandora rose to prominence a few years ago, but seems to be plagued by the same copyright and licensing limitations that the other systems had. There are BitTorrents, but these seem to be only good for popular or newly released albums.
The next best solution is music blogs. These sites are a great alternative to both Pandora and BitTorrents because they offer tracks that you might not have heard about (but might enjoy because of someone's suggestion) or that are not otherwise available for download. As best as I can determine, they operate on the edge of legality and quickly take their content off line if they are deemed to be infringing on some (ridiculous) copyright law.
A step up from the music blog is The Hype Machine, a music blog aggregator. It takes all these great blogs, lets you listen to the song before taking you to the place where you can download it. Rather than being a "walled-garden" like the above mentioned services, then entire internet is your playground when you use The Hype Machine.
While you've heard of bands that have made it big by way of MySpace, I've heard (mostly by word of mouth) that a number of bands are starting to come out of the music blog scene.
An example of this is the amount of excitement surrounding Parisian electro phenom Uffie. In anticipation of an upcoming album release, she's been releasing tracks to various music blogs for the past year or so.
Check it out.
(for my personal recommendation, shit doesn't get much hotter than Uffie's track "Pop The Glock")