I enjoy good singer/songwriters, and there's plenty of them in the older country, non-poppy/glitzy "New Country" genre. Merle Haggard and Guy Clark are two of them. I also enjoy Lyle Lovett. I admit I don't really look for new music to listen to in the country genre...
I really like bluegrass, too. Alison Krauss + Union Station are excellent musicians. I used to listen to Bela Fleck a lot and kinda got burned out. I wouldn't call his style exactly bluegrass, though. Seldom Scene is a local bluegrass band that put out a Grammy award-winning album last year that my wife and I really enjoy (called Scenechronized
I also listen to some Alt. Country: Ryan Adams, Ray LaMontagne, Avett Brothers, even recent Mark Knopfler falls in this category. One of my favorite recent CDs is a soundtrack collaboration btw. Benjamin Gibbard and Jay Farrar called One Fast Move of I'm Gone
All I can think about when I see the name Conway Twitty is how Family Guy
played a large portion of this video
clip in an episode -- perhaps more than one episode.