Lots of nice natural and "alpine style" trails in Cyprus. Dusty, rocky and loooooose during the summer, super grippy and fun after they've soaked a bit of water. The best of it is that you'll never need to use the words "sticky mud" over here ;-)
Trails are usually build and are regularly maintained by the local Forest Department. They are build as hiking trails but besides a few you'll very rarely see a hiker in there. Officially it's not allowed to ride in them but I know from first hand that the official policy of the Forest Department is to make a blind eye.
Besides those there are 3-4 downhill courses on the island and a few DH races. (next one on October 11th)
I'm a professional bike guide in Bikin'Cyprus Adventures (http://www.bikincyprusadventures.com/) and I can ensure you that there is AWESOME riding on the island!
If you are looking for company to regularly ride join these two groups on facebook: Storm Riders Cyprus Downhill