Just an FYI - there is not a birth center in Atlanta. There is a campaign to open one (google Atlanta Birth Center). N. Fulton hospital is trying to re-brand its Labor and Delivery unit as a birthing center, but it's still just a hospital L&D unit.
Alpharetta Women's Specialists attends deliveres at N. Fulton and accepts Medicaid. Dr. Joe Tate is an amazing OB (very midwife like) and attends deliveries at Emory Midtown. I believe Dr. Tate accepts Medicaid. SeeBaby (Dr. Brad Bootstaylor) is also beginning to offer OB services in addition to MFM. Not sure if he takes Medicaid. He will be delivering at Atlanta Medical Center, and may begin delivering at EUHM, too.
If you join the ICAN of Atlanta message board you can read lots of provider reviews.