Mountainbike on the trail or in town ? It makes a diference.
For the trail: try a double adjustable strap: one around your neck and one around your chest, both interconnected to each other.
In that way the camera bounces around a lot less.
Otherwise a case that fits the belt of your pants can be helpfull. I have this for my dP&S, there might be ones that fit a small SLR, and a F3 is not the smallest one.....
I would not fit the camera to the bike, esp on rough terrain, it will shake the screws out of it, unless you have one of those softtail all suspension bikes, like I have in Holland.......
If you want to mount it on your bike in town you can use a (super) clamp with a ball head and QR plate, it will require some experimenting.
Very sturdy super-clamps are made by Manfrotto(-Bogen in the US ???).