There are a lot of bands/performers in this thread that I would consider fairly well known, and even very well known. A number of them had pretty big hits at one point (Commander Cody, Asleep at the Wheel, James McMurtry, and a lot of the 60s bands listed). A bunch of others have had a lot of airplay like the Decembrists, although maybe on alternative stations. Others, like the Aristocrats, aren't very radio friendly (long, complex jams), but you run into them in music magazines and such. It's really interesting to see the groups people are listing. Lots of fun, and educational.
Here are a couple more. Chris Smither is an amazing bluesman and guitar player. He's written songs that others have made fairly famous, and performed with a lot of big names, especially Bonnie Raitt. I've seen him in concert multiple times, and he never disappoints.
And since someone posted the Decembrists, here's another alternative radio hit band that may not be on everyone's radar. I've seen them once, and I think they won't be "unknown" for long (if they even qualify now). The Lumineers.