I don't recommend taking this road...but that's just me. I think there are things you can do with out having to stoop to your neighbors level.
You and your neighbors could contact the city zoning board and see what you can do about the dirt track. It's not safe to have dirt bikes zooming around when children are playing nearby. You could speak to the City council and see if you could get something changed so that he could no longer legally have a dirt track in his backyard. It makes no sense to allow something like that in a residential area. They don't allow shooting ranges, archery practice and a number of other things because of safety concerns, it seems to me that this would fall in a similar category.
I grew up in a somewhat rural area, but lived in a pretty normal neighborhood. One of our neighbors decided to keep horses. It never bothered my family, but I guess the neighbors who bordered the family with the horses had issues with the smell and that the run off from the family cleaning the horse stalls would run into their yards near where their Children played. One of the families went to the city zoning board, and to a few town council meetings and they were successful in getting the neighborhood re-zoned so that barnyard animals could not be kept. I am sure there are zoning laws in place that don't allow things like dirt bike tracks in certain areas, it may not be as difficult as you think to get your area re-zoned. Sometimes neighborhoods are built up and just overlooked until someone brings it to the proper authority attention. It might be worth a shot.