Not too long ago a company called Mosaic Engineering surprised the DSLR world and came out with a filter that greatly reduced the aliasing and moire patterns on the Canon 5D Mark II. Installation was relatively straightforward, and the only major drawback was that super-wide lenses could appear very soft, especially in the corners. Now they've released a similar filter for the Canon 7D, and as you can see from the video embedded below, it will work in much the same way. They are also developing a filter for the Nikon D800, which has similar moire problems as the 5D Mark II, even though I haven't really noticed it too much in my testing.
Video is no longer available: vimeo.com/41491834
The VAF-5D2 filter, made for the 5D Mark II, has been available in their store for some time and will run you $385, and the new VAF-7D is also available at a price of $325. Compared to the price of the camera, the 5D filter is a slightly better value, but obviously if moire is something you're dealing with on a daily basis, the price isn't really an issue. In addition to the D800 filter, Mosaic Engineering also has a version in the works for the Canon T2i, Canon T3i, and Canon 60D. It might be a pain to have to install a filter, but it's a lot better than a moire pattern completely ruining your footage.
It should be noted that the filter is installed while the mirror is up, so you will only be able to take still photos in Live View mode. This modification does not affect anything else about your camera, and it can be removed fairly easily.