Lots of people start out cyclocross racing on MTBs. The major noticeable differences for cyclocross are that cross bikes are easier to shoulder and carry, and have lighter, thinner wheels. If you do more than a few races a year you will eventually want a cyclocross bike- lighter for lifting over barriers, and faster on typical cross courses. Don't spend much if any money trying to upgrade your hard rock, not really worth it. If you have a solid fork on it, that's good as you dont need suspension on a cross course which is not as rocky or technical as a XC race course or singletrack riding. Many beginning "C" races are won with people on MTBs. Try the sport out with what you have, see if you like it first. If you want to optimize your bike for little or no cost, consider the following:
1) Remove your bar ends (required for racing)- dont forget to plug the open end of your bars for safety
2) Reverse your brakes- rear on the left side, front on the right. This is to prevent you from doing an endo as you brake while dismounting. I reversed the brakes on all my bikes (road, cross, Tandem and MTB) so theres no learning curve, I never have to think about which bike I'm on.
3) If you have a suspension fork, replacing it it (downgrading, really) to a solid fork will lighten up your bike and make it easier to climb up short steep hills typical of a cyclocross course. ButiIf this is gonna cost you any significant money, then don't bother, not worth it, save your $$$ for a cross rig.
4) Make sure you can clip in and out of your pedals quickly, every time. Nothing like being stuck in your pedal as you smash into a barrier. Funny for onlookers, but bad for your image. The most popular pedals for cross are Time, Eggbeaters (my personal favorite) and (less so) high end SPDs. But don't spend money- make sure yours are well lubed, and consider keeping the tension on the low side.
5) Buy a cheap chaincatcher to prevent chainsuck. This happens frequently in cross races, especially when beginners drop their bikes hard after carrying, bouncing the chain off the chainrings.
6) Use the thinnest, lightest tires you can find. You don't need big knobs- light tread works best most of the time.
7) Remove your bottle cages so you can easily throw your bike on your shoulder. Cross races are short and usually in cool weather (or if you're lucky, fricken cold nasty rain or snow!). You don't need to drink during a 30 or 45 minute race, and it's too intense to reach for a bottle anyway, plus its just more weight to lift. You'll get used to filling up before the race, you'll be fine.
If you don't really ride much offroad, you might consider selling off the hardrock toward the purchase of a cyclocross bike, or even both your road bike and MTB for one cross bike. If you don't race on the road, a cross bike might be all you need for all your riding.
Read Simon Burney's Book on Cyclocross (available at Amazon.com) and check out this link for good beginner tips
You can also view some good cyclocross race videos here for free, watch for technique and get psyched for Fall! http://player.narrowstep.tv/?player=cycling
Now go practise some barriers.