Perhaps you were at the wrong place. There should be no problem getting a Qatari driving license if you have a US one. Make sure you are at the Traffic Department (not just the regular police station). Your employer should be able to tell you where to go. If your company does not have a mandoup who can show you what to do when you get there, I suggest you hire one of those guys who hangs out outside (make sure he has an ID!) and he should take you around and do what you have to do... You'll need to bring photos and a letter from your sponsor saying you are allowed to have a driving license. You'll take an eye test and fill out a bunch of paperwork. That should be about it!

It's been a while since I did it, so perhaps others can help you with the more up to date details.