Usually rental car companies will only accept credit cards. If your debit card is a VISA debit card, you should be OK.
Now while I logically would second what Grahame has to say, I have experience in driving with someone who learned to drive here--- with all the worst habits of Dominican drivers - ie. making a right hand turn from the left lane in front of another car, not stopping at intersections, driving too fast on highways, tailgating, leaning on the horn. I kept letting him drive because I reasoned that he needed the experience (which he clearly did!) and I kept trying to correct (gently, oh so gently) but finally, in Bani, I figured that it was just not worth the cost to my own health and he was going to have to learn by obsevation. So if the OP has learned to drive in the STates or anywhere a fuera, he may not be prepared, but he is probably a better driver. (My apologies to the Natives on the Board, I attribute it to your simnply not having had cars for that long and therefore not being taught by your fathers!)
:ermm: