Napster does offer the service you are describing. They've been offering it for a couple years now. I have a subscription myself. It's $10/month and I love it. I've shared it with friends who also love it. I can't use it on my work PC (Linux), but I still keep it because A) I have three other computers that can use it, and B) if I MUST listen to it while working, I can download tracks from any of those computers to up to 5 portable devices that support playsForSure. Almost all the music I have looked for is on there and most of it is included with the monthly subscription.
Once in a while you run into a stubborn label that wants you to pay for their music. It's probably only 5% of albums, so unless it's an album I'm dying to listen to, I just find something else. If I -am- dying to listen to it, I'll just buy the CD.
Napster -- in its current form -- is the only reason I haven't got a Zune (or an iPod, for that matter). And a lot of labels are on board. More than you may think. Not to sound like a salesman, but I am pretty sure they have a free trial, if you have a windows box.