Disclaimer - I'm not an expert, just had some experience with this issue...
Five tips to offer:
Try to ride with a firm, but relaxed grip. The firmness may well vary based on the difficulty of what you're riding, but try to make a conscious effort to relax your grip periodically on the trail where you deem it safe to do so.
We often ride with a finger or two on the brake lever depending on sight lines, downhilling, etc. I've found that 1-finger braking is much more comfortable for me than 2-finger (which I did for the first few riding years).
When you ride, your hand and forearm should be in alignment - in other words, the wrist should not be noticeably bent up or down as it puts a good deal of stress on the muscles in the lower arm and wrist. I was doing this for awhile a few years ago and I started getting ongoing wrist/forearm pain.
Try to evaluate whether you're "reaching" too far on your bike. This can be tough without an experienced eye helping out, but if the stem is too long, the seat is too far back, or the bars are too low, you could be putting an inordinate amount of your body weight on your hands.
Lastly, if none of the above works, try to borrow a (or buy a cheap used) pair of ergonomic grips like the Ergons (http://www.ergon-bike.com/us/en/home
). I find them to be very comfortable (providing you get the wrist angle set correctly - see above) and am contemplating replacing the grips on my brand-new bike with a pair.