Reduce the tension on the pedal some, especially until release becomes second nature. I've found that my pedals release better (and the motion is more natural) if I turn my heel inward instead of outward. You need to be careful not to stick your foot into the spokes but I don't have to turn my heel inward that much.

As others have said, learn slooooow speed technics. On a mountain bike, it's pretty easy to remain upright at a dead stop for several seconds. If you want to do this learn how to do it on level ground. With your feet on the pedals and parallel to the ground, slow to a stop while holding pressure on the front brake. I find that I do better if I'm standing on the pedals and not seated on the saddle. Turn the wheel slightly towards the crown of the hill or towards some other slight uphill. Put pressure against the brake with your forward facing foot (I prefer my left). If you feel that you are going to topple over, release some of the pressure on the brake and let the bike move a little forward (We're talking fractions of an inch here ) Keep practicing, you'll get the feel of it soon enough.

Once you can do it on the level, use the same technique as you are climbing. When you reach a point where you need to get off the bike (it happens to everyone, no matter what the yahoos tell you ), apply pressure to the front brake and your forward pedal and unclip the trailing pedal. It's best to practice with both feet, however because you might have to unclip on either side, depending on the trail.

Good luck