So long as the trailer mount/hitch attachs safely and you can stop/control the bike - shouldn't matter at all what kind of bike you use. I've ridden with a fixed conversion (road) bike, my regular road bike and my fully fred commuter MTB and they all work just fine for towing both my daughter and misc. cargo around in the trailer.

My MTB is by far the best suited bike for the job, but I know I can still use the others in a pinch - I feel faster with the road bike at the helm, and WAY more cautious when riding fixed (yes I run a brake).