Pay in advance online with a debit card. Check if your chosen hotel has this option, either on the hotel's own website or through an online booking service such as Orbitz or Hotwire. Paying the entire fee upfront has the added bonus that you may be able to get a better rate.
Bring the debit card you used along with you on your trip, as the hotel may want to put a small hold on it to cover potential expenses such as room service or phone calls.
Call your hotel ahead of time if you have not paid online and carefully explain your situation. While you will be unusual, it's probably not the first time the staff has encountered this and there's likely a procedure.
Ask if the hotel will accept cash or a debit card in lieu of a credit card. If you allow the hotel to put a hold on your debit card when you arrive to cover your stay, ensure you have enough cash in your checking account and have the hotel staff confirm to you exactly how much the hold will be for so you don't run out of money in your account to pay other bills.
Be prepared to leave a large cash deposit with the hotel when you arrive, if you have not gone the debit card route. Some hotels will ask for the deposit to cover the whole stay, so make sure you have enough cash with you.