@unesavage: Most likely there's a defect in one of the monitor's boards and it's dying with little that can be done (aside from contacting the company/filing a warranty claim). That said... there's a very rare chance that something else is causing the issue (like malfunction with the video card) so before writing the monitor off I'd suggest doing a few simple tests, if possible.
The easiest is to test the monitor in the absence of a video signal from the computer and to examine the cables. You can unplug the video cable from the monitor (but leave the power on) and cycle through the various on-screen menus to see if the line persists in the absence of a video signal from the computer. You should also take this time to examine both ends of the video cable and make sure there isn't any damage (like bent or loose pins). Also, if you have the ability to connect it with an alternative input, now would be a great time to try it out. If you're using VGA, try DVI (or HDMI if available).
While you're fiddling around in the on-screen menu system of the monitor, look for a factory reset option. It's a long shot, but perhaps resetting the monitor will resolve the problem.
For more advanced troubleshooting you can also hook the monitor up to another computer and see if the lines persist. If they do, the defect is in the monitor. If they do not (and the monitor appears fine when it's in test mode and/or hooked up to a new machine), then there is something strange going on with the original computer and you'll need to find another monitor to test the original machine with to see what's going on.
Finally, if you're willing to invest some time in order to stave off spending the money on a replacement monitor, don't rule out opening your computer case to clean the GPU (blow the dust out, make sure the fans are operational and there isn't dust caked on the blades). Stressed/overheated GPUs can do quirky things and vertical/horizontal artifacting isn't out of the question.