West Virginia - The Real McCoys (Not sure how many viewers in 1957 - I may have to drink. Dang.)
Actually, they were only from WV. The show was set in CA. BARTENDER! Another drink if you please. 
I cannot remember the name or the state… but the sitcom with John Ritter and Billy Bob Thornton was about a Southern state senator if memory serves… I want to say Arkansas.
Colorado - South Park
Er… um… - The Simpsons
Bonanza takes place in Nevada. The Ponderosa is near Virginia City.
I looked for something more mainstream but had no luck, so:
Tennessee – Hee Haw (or The Grand Ole Opry)
Indiana:
One Day At A Time has been mentioned.
In this last year, there was a police show set in Indianapolis. I don’t know the name. Neither show was actually filmed in Indy, except for a few stock street shots.
A Christmas Story, shown every year, was from Jean Shepherd’s stories, which were always set in northern Indiana.
Another made-for-TV movie was set in Vevay, IN. It was about a little girl rescued from very primitive conditions (she didn’t know what toilet paper was for.) Cloris Leachman played a grouchy backwoods crone.
For Gold and Glory was a documentary about a black auto racing circuit, mostly based in Indiana.
Don’t worry about the drinks. Believe me, I’m keeping up.
Newhart has been mentioned, but The Bob Newhart Show was set in Illinois, and Newhart’s last episode ended there.
The new comedy Free Ride takes place in Missouri
Georgia: Dukes of Hazzard, Carter Country, Designing Women
Here’s a couple I’m really surprised no one jumped on right away, given who frequents this board:
Kansas: Smallville
New Mexico: Roswell
And a few more, courtesy of IMDB Power Search:
Alabama: Any Day Now
Connecticut: Judging Amy, Who’s the Boss?
Mississippi - In the Heat of the Night
A good question may be: is there a state that has had no (networked) TV show about it?
Virginia–The Waltons
Minnesota–Little House on the Prairie
Florida–Surfside Six, Penacola: Wings of Gold, CSI Miami, Nip/Tuck, Everglades, Flipper, I Dream of Jeannie, Gentle Ben.
An oldie for Kentucky–Daniel Boone!
I’m pretty sure Montana fits that bill. I wouldn’t lay money on it, but I cannot think of any TV shows set in Big Sky Country.
Nevada - Reno 911!, CSI: Las Vegas
-Otanx
Missouri has Grace Under Fire I think.
Mississippi: In the Heat of the Night***
TV series actually filmed south of Atlanta, but set in “Sparta, MS.”
Aha! I found one for Montana! According to this My Friend Flicka (1956-1958) was set there on the Goose Bar Ranch.
Can’t seem to find anything for Idaho or North Dakota though.