Well, the idea of REM is it is the deepest sleep your body will allow. This is different from person to person. So and hour of REM sleep is worth more than an hour of none REM sleep. If you are a really light sleeper REM is less restful than if you are a really deep sleeper.
I, however, am an extremely light sleeper. The slightest sound or a change in temperature can wake me up. So my REM cycle is not much more gratifying than if I was in a non REM sleep. It takes the average person, I have been told, somewhere between 1 and 3 hours of sleep to start up REM, but the deeper you sleep the quicker you start REM.
I wake up a lot because I sleep so light. I usually wake up every hour or half an hour that I sleep. So for me to start up REM is very difficult. And, again because I sleep so lightly, I don't really require REM to be awake and healthy, it just would really help.
So in reality, if you sleep heavily, then yes, you pretty much require REM. If you sleep light, no, you shouldn't.