If you're talking about Dhahran, in the past I have seen quite a few people dirt-biking in the area between Peninsula Boulevard and the outside perimeter fence, an area roughly bordered by Haradh St, Canyon Road and the Peninsula soccer fields/spray fields. I'm not sure about places outside camp to do this, unless you count dune-riding as dirt-biking, in which case the dunes at Half Moon bay seem to be a popular area. Although the last time I was there, it was pretty crowded and I saw a lot more 4-wheel drives and ATV's than motor bikes.
If you go to Half Moon Bay area, you go early in the morning. Once the locals arrive it can be dangerous on a motorcycle, some of them think its funny to drive their vehicles close to you at high speed to try to spray you with sand. Happened last weekend to us. We left quickly, but had a great time earlier with no one around.