There is a lot of trend, fad, and overall misconception in the fitness industry and this buzz word, "core", is probably the most prevalent. Certainly trying to 'target' certain muscle groups in the midsection is an important component of training (especially if there is imbalance) but the fact of the matter is that the body is a system and, as martial artists, most will be using it as such. Train it that way too. Standing on one leg on a BOSU ball with your arms outstretched with a purple dumbell in one hand and a swiss ball in the other while blindfolded doesn't need to be the way you train your "core." In fact, it's pretty ridiculous. Stick to big supportive movements with a few quality specialty exercises and you will be good. If you enjoy pilates or yoga or whichever and find it effective go for it, but don't think for a second that in order to "train your core" you need to do all this "balance/stability" training in magazines and infommercials.
For "training the core" I would recommend:
Planks
Ab Wheel (not the crappy assisted one that does the work for you)
Weighted side bends
What I wouldn't recommend:
Stability ball - simply because most people do it incredibly wrong or just don't know what to do in the first place.
What will give you the most bang for your buck:
Squatting
Pressing overhead
Pulling (Deadlifting, Cleaning, etc.)
KB's.
You move big in your art, so move big in your training.
Also, it might vary some from textbook to textbook but what I consider the "core" includes the hip flexors, rectus abdominus, spinal erectors, transverse abdominus, latissimus dorsi, Obliques, etc., etc.