To answer the original question, I don't think the cambios (except maybe FX Trader) take debit cards.

Cash always works, travelers checks might work but you should expect a lower exchange rate, and anything you can use at an ATM works.

When you call your card provider to tell them you'll be in Jamaica so not to issue security holds on your card, also ask them if they can bump the daily withdrawal limit on your card. I had mine bumped from $300 US to $500 US and it made things easier. The ATM itself may have a transaction limit, but that way you can at least do another transaction.

This may be obvious, but if you're at an ATM getting Jamaican money, remember that if you ask for $300 you'll get $300 JA, which is $3.30 US and will pay the same transaction fee as if you ask for $30,000.