The Cast Showoff: Real-life blues singer Chris Thomas King plays Tommy, and at one point gets to sing (in his own voice) a rendition of Skip James' "Hard Time Killing Floor Blues." Also, that's really Tim Blake Nelson singing the lead vocal on "In The Jailhouse Now".
Clooney did work really hard on practicing the song with a vocal coach and reportedly wasn't terrible. It just wasn't good enough for the movie and it was dubbed over.
Dan Tyminski had worked with Clooney on the latter's singing voice, and reportedly remarked that the little time they had available for voice training, wasn't enough to get Clooney's singing voice to the standard required for a #1 hit.