Some rental car companies like GOLDCAR will accept a Debit Card but you must have enough funds on the Debit Card to cover the Excess insurance amount. This Excess amount can vary wildly between 300-3000€ in certain countries. The Excess amount will be withdrawn from your debit card when you arrive.
GOLDCAR will accept Debit Cards on this basis in Cyprus, Croatia, Spain, Portugal, France, Greece (except Rhodes), Malta, Netherlands, Romania & Mexico. In Italy: Olbia & Cagliari.