Your best option is to practice a lot on how to unclip your foot, so you get accustomed to it. I suppose you have already done this. So you must now adjust the spring tension on your pedals. Refer to the instruction manual on how to adjust it.
Aside from that, I have found that dirt, grime and other pollution in the cleats makes it extra hard to get the foot free. It gets to your pedals every time you step on mud or soft soil and then go back to he pedal, something that does not happen so frequently for a road biker.
When you cleat after stepping on mud or soft soil your shoe picks up little stones, little stems, leaves and all sort of things that clog up the pedal's mechanism, and the pedaling action makes all this dirt get to critical places.
That's why I recommend to thoroughly clean the pedals and oiling them between rides. Normally you won't get too much dirt in only one ride, but you don't want it to get accumulated, for this worsens the performance of the release mechanism. Oil helps a lot for clipping in and out, but also helps a little by not allowing some dirt to stick to the mechanism (Your pedal won't be so dirty by the end of the ride).
Tip #2: you should periodically inspect your pedals and cleats for dents, deformations and other that can affect your ability to clip out. Specifically I've had trouble with the "ramps" that push the cleat out when you twist your foot. For some reason these develop a kind of hook that prevents the foot from twisting. You can easily file out these deformations, usually they are surprisingly tiny! Use a fine file or a rotary tool (dremel) with a fine grit grinding stone.
Tip #3: Part of some training I also did, was about conquering difficult technical sections without removing any foot from its pedal. I did this by building small obstacles in the middle of an ample flat terrain. The Obstacles where made of stones and logs, trying to make then with increased difficulty every time, but always made then in such a way that if I fell it wouldn't hurt much, hence the flat clean terrain around the obstacle. This kind of training gives the rider a lot of self confidence and builds the necessary skills that make the use of cleated pedals more intuitive at the required level for MTB riding.
These tips have kept me clipped for 10 years of mountain biking, doing XC, and a little all-mountain and downhill.
My final comment: IF you're accustomed to SPD on your road bike, make an effort to stay on SPD for your mountain bike, if you remove them, almost sure you'll miss them!