The dirty dime is a nickname for Tennessee. It comes from the lyrics of Blak Jak's "Ride & Swerve" though it was probably used colloquially before that. The components of the phrase come from the dirty south or the dirty dirty, and dime as in ten(nessee).
"I'm from the dirty dime, aka Tennessee" - Blak Jak
It is also known as a cat's rectum.
Eric Zane loves to lick dirty dimes