The "War of Northern Aggression"? ;)
There's a guy named Wayne Erbsen who puts out his own tapes, usually for sale at the gift shops at battle sites. His style is goofy, but if you're looking for songs to learn, you can get the tune and words. I think he's also a reenactor and, at least on his tapes that I've seen so far, it's all period tunes (as opposed to modern ones like "Two Brothers" and "Gray-Coat Soldiers").
"Wildwood Flower"--this was a top seller in sheet music in 1860, long before the Carter Family made it famous. I've also been asked for "Marching Through Georgia", "Garry Owen", "Minstrel Boy", "Cumberland Gap", "Vacant Chair".
Please, NO "Ashokan Farewell"!! Yeesh.
My brother has been into Civil War reenacting for 8 years (since he was 12). The men in his unit don't play much music themselves but I used to sit around with the women and play the lap dulcimer--my brother got mad at me once for attracting more attention than the soldiers did (probably because I was less scary to the kids). I've played with harmonicas and a few banjos, and Abraham Lincoln seems to be a particularly popular character for limberjacks.