Mountain bike riding is usually permitted on any public trail in France (do check for local exceptions). These trails are best found using the French IGN 1:25,000 map series TOP RANDO.
Most local bike clubs will organize at least one annual 'randonnee', which is a non-competitive ride on the best of their local trails. These usually cost 5-10euro, and include refreshments (start, finish and usually at least one en-route), and marked routes to follow. Search for "randonee vtt" to find events near you (VTT is the French for MTB, and stands for "velo tout terrain").
This ride hits many highlights of the Rambouillet and Dourdan forests, with long sections of singletrack linked by doubletrack. There is nothing too technical but it a nice rolling route with a fair bit of elevation gain over its length.