Two 19th century slave cabins still exist a the Point of Pines Plantation on Edisto Island, South Carolina. One of them is being painstakingly dismantled and sent to the Smithsonian Museum to be a centerpiece of their new museum, the National Museum of African American History and Culture, which is slated to open in 2015. Because the Union troops seized many of the South Carolina sea islands not long after the Civil War began in 1861, the enslaved people at Point of Pines Plantation were some of the first in the South to declare themselves free. There is a great article on this slave cabin over at the History Blog.
Other slave cabins that I know that still exist are at Bratton Plantation In York county, SC (these are brick and hopefully are not reconstructions, they look old though) and at the Bellamy Mansion in Wilmington, NC (I’ve actually done archaeological work on these). Their website has a video about their restoration.
addendum after further reading of their website I found that the Bratton Plantation slave cabins are reconstructions. They are done very well in my opinion, but not the originals.