I you have international roaming enabled on your phone, it will work in Punta cana.
If your phone is CDMA phone, it will work via Claro, Tricom or Centennial networks.
If your phone is GSM phone (witha SIM card - chip) it will work via Claro or Orange networks.
If your phone is CDMA phone, you can also get a second number activated on it (NUM2) at Claro dealerships. make sure they program it as a secondary number, so as not to lose your AT&T number and service!
If your phone is GSM phone, and it is unlocked, you can activate a SIM card with Orange or Claro (provided that Claro accepts passports for SIM activations already). Orange activation is striaghtforward, with your passport, and 10 dollars (300 pesos) you get your prepaid startup kit in 10 minutes. If your phone is locked, ask for an unlock code your AT&T service center or call center before you leave. Eventually, you can unlock your phone in PUJ after you come here. Just go to any cellular store NOT marked Orange (venta y reparacion de celulares).
You can also buy new phone at Orange, with a prepaid SIM card, for under 1000 pesos (30 dollars) or with Claro dealers you can buy used phones for as little as 300 pesos (10 dollars).
It is cheaper to use local service to call USA than your AT&T service. You pay US$0.3 per minute to call USA on local service (Claro/Orange), compared to up to US$3 per-minute roaming charges with AT&T.