Gurf was Lucinda Williams' original guitarist/bandleader/producer, and is great on his own as well. He is playing and producing around Austin now.
Buddy Miller, the best bandleader/guitarist/singer/producer in the world currently---a real MVP! Makes great records, with and without his wife Julie, as well as Emmylou Harris.
Someone already mentioned them, but surely the best and most influential Band who can be called A-C has to be The Band, who were an alternative to the whole musical and cultural climate of the late 60's, showing the way out of that mess. And we would be remiss in not reminding ourselves that Johnny Cash and Jerry Lee Lewis are about as Country as you can get, and were very alternative to Nashville for long periods of time. Okay, they're not Bands, but.....