Credit checks are allowed for debit card users although use of them varies a lot by location.
That said, I would NEVER use a debit card to rent a car, especially one that is tied to your main source of funds. If something happens to the car (ie. damage), Hertz can (and most likely will) take the funds right out of the debit account (you give them permission to do this when you rent) to fix the car/damage (yes, insurance/coverages should apply but while this is worked out, Hertz will probably assess the damage directly).
I can also give you a more personal example of an issue with a debit card. My sister-in-law rented a car last September and thought she was doing a good thing by using her debit card (she didn't ask my opinion) and she went to withdraw money from her debit card 2 days later and it said insufficient funds for a fairly small withdrawal. Well after checking her online account, she found out that for some reason the Hertz system kept processing holds for the rental on her debit card, locking out her entire account balance (around $2,000). She immediately called Hertz who realized there was a problem, agreed to fix it, but explained it would take 3 - 5 days for the funds to become freed up, fortunately she could just transfer other money and she was with all of us. However, this could have been very problematic if she had paid her mortgage just before she left and this happened her check/transfer could have bounced, etc.