My DH suffers from this. I think it's hereditary, as his father has the same problem.
The experts have recommended that you DON'T just lay in bed. When you wake up and don't go right back to sleep, get out of bed, and do something soothing/relaxing until you feel sleepy, then go back to bed. You can get a glass of warm milk, read, listen to quiet music, etc. The idea is that your bed should only be associated with sleep (and other things that are bedroom specific). You shouldn't watch tv there, or read or eat, etc.
Hope it works out!