Yes, you actually can, no matter what some people may tell you. Depending on if you're planning to use an iPhone 4, (I'm guessing that you are, since they can now be factory unlocked) you might need a MicroSIM or you may have to trim your current SIM card down to scale to fit the smaller slot on the iPhone 4.
Calling and text messaging can work fine on a prepaid iPhone, but if you want to use data, that's the hard part. There's a special process that you must go through from a website that can get prepaid data for the iPhone working for you. An alternative is just to use the phone's built in Wi-Fi.
where are you on your contract? Part of that is paying for the phone equipment. Before you could switch you would have to pay the termination fee which probably varies according to how many months you have already paid for. If your original contract is through then you could get a prepay option.