Good technical photographical comments by Rob.
I'll add some of my own experience.
I'll start off by saying that action photography is HARD.
I do a lot of frame-grabbing from video, and I've found that the most interesting shots aren't what I initially expected. The essence seems to be capturing the human response, be it intent or reaction. For martial arts, the best shots aren't when the strike lands.
They seem to be the starting point of the actual attack, which is the slight pause between the windup and the delivery, using baseball terms. This is not a uniform rule,
but simply a generalization of my observations. Sports photographers should really be rifling off pictures in burst mode, which only the latest crop of prosumer cameras can handle. Video rates would be ideal, but the resolution is currently too low.
Most of the comments below are directed towards indoor photography, which is where American martial artists generally train. I envy those lucky enough to train outdoors.
I experienced the same disappointment with my digital camera action shots - generally blurry. The video feature proved to be FAR more useful, although the video frames are too low in resolution.
So, I spent a lot of money on better gear. I bought
a camera with a detachable lens so that I can add my
own indoor lens, one that opens wide enough and still
allows a fast shutter time. Most consumer digital cameras have a 1/20 to 1/50 of a second shutter time
which isn't fast enough to freeze the action.
Even with the good gear, a 50mm f/1.8 lens will yield poor picture quality unless you use a flash.
Here are some tips that have helped me take better pictures for indoor events that have horrible lighting,
horrible at least for digital cameras.
1) Use a higher ISO value, such as 400-800
This boosts the picture level at the expense of noise,
but you will be able to tell if you have a decent exposure in the LCD review panel.
2) Use support to reduce user camera shake
I use a monopod, which is a one-legged support. It
is far more portable than a tripod, so I end up using it more. It doesn't remove all camera shake, but it will
reduce it substantially. A tripod is ideal, but that usually makes for boring shots, in my opinion. The shots
are boring due to the lack of variety in camera angles, I believe, even with the improved clarity. If you don't have either, lean against a wall or tuck in your elbows.
3) Don't press the camera shutter button too hard.
Many people stab at the button, causing the camera to move during the exposure and causing more blur.
4) Use continuous auto-focus when the subjects are
all over the place.
Many cameras lock-into focus before snapping the picture. This is a contributor to the delay between the
button press and the exposure. By then, your subject is
out of focus anyway. Sometimes, it's just better to manual focus at a particular distance and stay there as autofocus will mess your shot anyway by focusing on the wrong object, often the wall behind your subjects.
5) If you use Autofocus, make sure the zone is on the
face of your subject if you're doing a portrait. When the eyes are in focus, the picture is good. Otherwise, focus
on a high contrast part of your subject even if it messes up the framing of your shot. At least your subject will be in focus.
6) Take a lot of photos when the subjects aren't moving.
These aren't exactly action shots, but you'll be surprised
at how many of these you want to keep.
7) Buy the biggest, fastest flash card you can afford,
and an extra battery.
Take tons of pictures without worrying about filling up the card. You will have many more junk photos, but you'll also have more keepers.

Use available light
If the light is coming from one side, position yourself
to take advantage of it.
9) Use manual mode to adjust shutter speed versus aperture.
With many cameras, this leaves you with the choice of
dark but frozen action or bright and blurry action.
You can always add motion blur in Photoshop .
10) Shoot in RAW mode.
This is the highest resolution format for your camera.
JPEG is ok, but if you really need a good picture, you might get all of the data. Often, the camera will over-sharpen the entire picture, adding a lot of noise
11) Learn to use Photoshop.
It's amazing how many pictures can be rescued by Photoshop processing. The Crop tool can help you turn
an ordinary picture into an outstanding one.
12) Take a picture of a white piece of paper
It will help you with White Balancing after you've left the scene.
13) Don't zoom, get close
Zooming on most consumer cameras reduces the brightness of the lens as it can not open as wide at
full zoom. This means the shutter speed will drop.
Also, at zoom, you'll have more camera shake . On the
downside, your subjects will also move more because they occupy a larger part of the exposure.
Wow, I didn't think I'd write an essay. Don't get me started, ok ?
Hope this helps,
Gints