Friend of mine had the same situation.
She went through Select Qatar. They rent short and long term. If you pay the entire amount up front (I suggest going for 3 months), there is no need for a credit card.

But, like someone suggested... suggest getting a driver... even if just for the first week or 2.... it will make a world of difference just in getting a sense of direction. its difficult enough to navigate between white landcruisers and crazy buses and taxis without knowing where you're going.

Good luck!