I finally created an account just to answer this question. The answer is unfortunately that Americans now have to pass a driving test in order to get a license. The explanation I got was that in April Obama passed a law that now requires Qatari's to pass an American driving test in the US, so they reciprocated the punishment. I personally just went through the whole procedure last month after my wife got her license in Feb with no hassles.
You must go to the driver's license office first and try to get a license. At the very end of the entire procedure someone will ask for your driving test results. Of course you won't have it, but then they will give you all the required documentation for the test. They also tried to tell me I needed to take a 25 session class for thousands of riyals. I found out later that its not required, just a passed test. You must have this documentation when visiting the driving school of your choice in order to schedule your driving session.

I used the school just north of Landmark Shopping Center. They were very nice to me, and really like Americans for some reason. I called the school to schedule a test and they told me to come right in, so I did. However, when I got there, they only scheduled the test for another day and collected 50 riyals for my effort.

The test itself starts at 5:30am and takes about 4 hours to get out, even though there was only about 10 minutes worth of driving. You have to wait for everyone to finish each section before moving on to the next and finally finding out if you passed. I had to pay a rental fee of 100 riyals for the test car.

The first test is the road sign test. STUDY the British road signs and roundabouts. Americans WILL FAIL going in blind. I found a great website that will help. http://www.ukmotorists.com/

The next test is VERY technical and tricky! There were about 150 people there to take the test. About 50% of the students had taken the class to prepare. Luckily I made friends with another American that had taken the 25 session class and still failed the 3 times he took the test before (he failed the 4th time too). He told me some of the procedures and tricks you learn in the class but never get told without. They basically put you in an old beat up car and expect you to know what to do on a crazy obsacle course. The only tip I can give you is that want to pay attention to everyone else in detail and don't get in line for a car until you think you know what to do. PS, all cars have MANUAL transmissions!

The last challenge is the road test. My test was very short. About 2 kilometers total with 3 roundabouts to negotiate. Study the british rules and be smooth and confident. Use turn signals, seatbelt, adjust mirrors, adjust seat. This is by far the easiest test of the 3.

At the end of the day, I managed to pass. I had to pay another payment of 50 riyals I think, to process my license, but at least I never had to return to the drivers license department to finish the procedure. I was one of 12 from 150 who did pass. I was the only one who didn't take the 25 course class. I was very lucky. Bottom line is that traffic schools in Doha are there to make money, and they can make a lot of money by not passing people.

Good luck and try not to get frustrated during the ordeal.