Jon, at the indoor range I go to in Albuquerque, they allow AK-47s and AR-15s, even with FMJ bullets. Rapid fire? No problem. The even rent full auto weapons for only about $20 per day. (Although you have to buy your ammo from the range shop, and a full auto can eat a lot of ammo very quickly.)
I've not shot an AK-47, although the ones I've handled sure seemed cheap with all the stamped metal parts. I know they're virtually indestructible, but I still prefer an AR-15.