Johnyboyd - I recently started a thread a mile long about the horrible eye strain i was experiencing.
A lot of the sets you see in the store are set up in such a way to look better in a store environment, and are screaming for your attention. This means the brightness and contrast are cranked up to their maximums, which is the #1 cause for eye strain.
Outside of that, at home I still was experiencing unbearable strain even when the brightness/contrast were at appropriate settings. Some of the picture pre-setting modes, such as "pro" "movie" etc contain certain sharpness/enhancements that alter the picture to make it look "better." Unfortunately these modes made the display VERY hard to watch. I'm not quite sure why, but with me i noticed the more of the enhancement and junk i got off the screen, the better i was able to watch.
Another important factor. I found that certain colors are easier on the eyes than others - meaning that if the colors weren't balanced correctly one color may dominate and make the screen a little bit harder on your eyes. It's also good if the color temperature is right for this as well.
I don't know the science of why certain things affect the eyes more, nor am i going to make it up, but i do know I have sensitive eyes which means my televisions have to be just right.
Backlighting reduces eye strain in that your eyes are constantly adjusted to it's neutral light output. Movies and television shows often change very quickly from light to dark scenes, and your pupils are contstantly trying to adjust, which tire the muscles. If you have a neutral/bias light source behind your tv, your eye muscles will have much less a hassle running between extreme lights and darks coming from the display. I didn't like the subtle backlighting effect at all, as i thouht it ruined the movie-in-the-dark experience for me, but not everyone agrees with this.
With moving objects giving you headaches etc, viewing distance might be part of this. Seating for HDTV is typically 2 times the screen width, give or take a little.
But I still think if your getting eye strain from even seeing a display on a showroom floor, then what i mentioned above should be the solution. Getting the colors right, getting the brightness/contrast right, and eliminating extraneous edge-enhancements (so that dvd's and broadcasts are in their true/original form) will make your set much easier on the eyes.
The long and short of it, sets NEED to be calibrated in a way that's industry standard yet pleasing to your eyes as well. If you have the knowledge and experience, you can definitely do much of this yourself, otherwise it's sometimes beneficial to pay the 250$ for a professional to do it.