Everything that doesn't move quickly looks fine, so applications look fine for the most part. The exception being web browsers when I scroll through the tabs quickly. I believe the content from the 360 is in 1080p, although I have seen that some games are upscaled from 720p.
I was trying to see if I could reproduce the behavior consistently last night and the only thing that seems to cause the effect is fast motion, but the effect doesn't seem to consistently present itself during fast motion.
After doing some more research into the screen door effect, I think that might be a possibility. I will try to adjust the focus when I get home and see if that helps at all, that has been the only "fix" I have read about for the screen door effect.
If anyone has any suggestions on anything else to try I would be grateful. If not, suggestions on new monitors would be helpful as well, since I don't plan on living with this effect.