There are no birth centers in the Atlanta area. In fact, the only birth center in all of Georgia is in Rincon, which is an hour outside of Savannah.
There is a birth center in Greenville, SC, and a couple of birth centers in Tennessee. If you are really looking for a birth center, it might be worth considering the drive to one of these. Greenville isn't that far.
There is always the homebirth option. There are several homebirth midwives in this area.
The most natural birth friendly hospital in the Atlanta area is North Fulton Regional Hospital in Roswell, with the midwives at Northpointe OB/GYN
www.npobgyn.com they have a waterbirth option there.
It is, however, still a hospital, with hospital policies, and their backup obs are very traditional. I wouldn't call it "birth center-like" but as far as hospitals go it is OK.
Good luck-
take care