To explain, the NTSC video signal works by drawing 525 lines of video on the screen. However, rather than drawing all 525 lines in a row (something called progressive scanning), it first draws the even lines, then the odd lines; this is interlaced video, and each group of lines is called a field. The interlacing is controlled by a small timing offset in the signal so that the odd fields start a little later.
Older video game systems tended to only support about 240 lines of information, so when they output a video signal, they would just output fields without doing the interlacing offset. This works fine on analog televisions. However, the with the LCD TV, you're not directly seeing the video signal. This LCD is a digital display with a native resolution of 800x600. It can't be directly illuminated by the video signal; instead, a computer chip in the LCD TV reads the video signal, digitizes that to a memory buffer, and then displays that memory region on the LCD.
The problem here is with the digitizing process. The set is handling interlaced video without a problem; each field fills alternating rows of memory, and when each frame is complete, it displays that on the screen. However, with a video game providing only the even rows, the digitizing process malfunctions, and it doesn't show a complete picture.
My Dreamcast starts with interlaced video with the boot screen, but when I started a game, it went into non-interlaced mode, and then I only saw half of a picture, with alternating black lines running through the image. With my Nintendo 64 and Super Nintendo, the timing of the video signal was odd enough to put the TV in a mode where I'm saw the first image the game system output interleaved with what was being currently output. Either way, the display is useless for running those systems.
When I called V Inc. technical support earlier, they didn't seem to know of any problems, but I'm pretty sure that I wasn't talking with a video engineer. I'm going to keep the set -- it works great for my primary use, and I might have a way to get video games working with it using some extra hardware I've got lying around. However, if you're in the market for a LCD TV to play older video games, be aware of this issue and be sure you can test it out and return it if it doesn't work.
(Update: A good article on NTSC video signals and why interlacing is part of the standard is here on the Gamasutra site. You need a login to read it; the site is mainly for people in the video game industry.)