How nice that you are planning a road-trip in Ireland! From what I know, a credit card seems mandatory with most rental agencies, unless you have an Irish issued driving license.
I would suggest getting in touch with your bank prior to your trip. Some companies may allow debit cards but you would have to pay a higher insurance premium or a significantly higher deposit.
Feel free to get in touch with the Car Rental Council
or have a look at these few tips
on hiring a car in Ireland for more information.
Other than that, have you already figured out a a road-trip itinerary?