It really depends on the type of course. If it's flat and fast then you're s.o.l.. If there's a lot of gnarly singletrack, and steep but rideable climbs, then you might do alright. In general, however, the courses are such that a cross bike is the optimal tool.
The chief disadvantages of a mountain bike in a cross race are the weight, the small main triangle, which hinders shouldering, and the non-aerodynamic riding position.
I would say to go ahead and try the race on your mountain bike. A cross bike makes the race a bit easier, and arguably a lot more fun, but even with a mountain bike, you'll know after your first race whether or not this is the sport for you.
I remember a few years back seeing some guy racing the C races with an old boat anchor steel mountain bike. He totally decimated the competition. Left them way behind.