Jim,
They have a small area of zone 8a-8b near the border of Turkey and Greece. They should share some common varieties with northern Turkey and Greece, to my understanding. However, I do not know how those varieties would do in the US because their climate is much drier, and they do not have much heath, by American standards. Germplasm repository in CA has three pomegranate varieties that have originated in Bulgaria: Nikitskii Rannii, Shrin Zigar, and Hotuni Zigar.