I have towed two out of three of my boys in our Burley D'Lite (the toddler will be ready for it this summer). I always towed it with a road bike until last summer when I bought a mountain bike. Although, the mountain bike felt steadier, the road bike was fine to begin with. So, the mountain bike was an improvement, but not one that made the road bike worse, just different.

As a related aside, when my oldest got too big to fit in the trailer with the middle son, I switched him to a trail-along that connected to my seat post, then attached the trailer to the trail-along. The "bike train" (as the boys call it) DEFINITELY tracked steadier with the mountain bike than with the road bike. But, that configuration is far different from a single trailer no matter how heavily loaded it may be.