The Great Jura Crossing, La Grande Traversée du Jura, is France's premier long distance mountain bike trek. It takes at least ten days, going along the Jura Range which is an uplifted limestone plateau that forms the border between Switzerland and France to the north of Geneva.
The tail starts near the city of Montbéliard and travels south down the Jura Range to Culoz which midway between Lyon and Geneva. The Jura plateau does not have the jagged high peaks of the Alps but it is undulating and the trails are sometimes rough so this is a challenging bike trek. The scenery is often stunning although you won't find the majestic beauty of the Alps, and there is plenty of heritage, culture and rural charm.
The route is a combination of back country roads, forestry trails, farm tracks and walkways. Sometimes the going is easy; at others it's a real bitch. You need at least intermediate mountain bike skills to enjoy this ride. It's usually done as a supported ride with a tour operator but you can do it independently, staying in lodges, B&Bs, hotels and so on. Don't even think about carrying a tent and camping gear - the tracks are steep at times, rough at others, sometimes both. You need to keep the weight down.
Navigation can be difficult. There are lots of signs but you can't rely on them. Download a GPS trace of the route before you set out. You can get it here: http://www.gtj.asso.fr/fr/differentes-formules-de-la-gtj/gtj-a-vtt/les-donnees-gps.html. If you haven't got a GPS use your smart phone with an app such as Back Country Navigator.
Your base for the expedition should be Lyon. It has an airport, good train connections and is a great city. You can store your surplus gear in your accommodation till you get back and hire bikes at "Location Lyon", http://www.lyonlocation.fr. Catch a train to Montbeliard (not the fast train) which is just a short ride from the trail start at Mandeure. There are good connections back to Lyon from Culoz where the trail ends. And there are numerous places you can exit the trail if things go wrong.
The trail has a website here: http://www.gtj.asso.fr with lots of good information and photos.