Mountain biking

In Spessart there is a well-signed network of mountain bike routes, 'Bikewald'. To be honest, I was a bit dissapointed with them. The easier (blue) routes are very nice if you have children with you, for example route 33 starting from Gräfendorf both has impressive scenery and they have made a good effort to include some technically difficult parts (which are short enough that children can walk up).  But the more difficult routes (red and black) are mainly difficult for having a lot of hills, some of them steep, but unfortunately a huge proportion of the route is on gravelly fire-road through the woods.  Those are mostly very broad, with a horrid gravel surface to cycle on and often going on in a straight line for many kilometers. 
Klaas Bergfeld's site about north Spessart (www.bike-park-nordspessart.de) on the other hand has some really nice routes.  A lot of singletrack, and when they are forest roads, not the great wide motorways of the official route (though that might be due in part of different forestry managment policies of Bavaria (Bayern) compared to Hessen).  I had some great rides following his tracks and can thoroughly recommend them