What console are you planning to get?
Your LCD is a native resolution of 1600x1200 in 4:3 aspect and has DVI and VGA inputs. The few reviews I saw indicated, like most LCDs, it is not very good at non-native resolutions. If it doesn't have pixel mapping you may have some issues. With pixel mapping you get letterboxing, so a 720p image would show up with black boarders (160 pixels on each side, 240 top and bottom). Without it a 720p image would look pretty poor in most cases. Now some LCD HDTV Monitors (Monitors with TV support) have better scalers and support Component and so forth, but yours does not.
So the weight displaying the images correctly will fall to the console in most cases. The 360 has officially supported VGA cables and they work flawlessly in almost all games due to the internal scalar. You will need to get a response from a forum member to see if it supports a 1600x1200 resolution though as scaled output. Not sure it does. As for the PS3, it currently doesn't have the scaler being used and would need to find someone who has tested not only some sort of conversion to VGA/DVI, but also on a set resolution of 1600x1200 if the dashboard even supports it.
That is why I plan to use a CRT. You can get big ones for cheap and you have more control over the image and image quality. And I have a 21" laying around. 720p works fine on these (ditto 19"), the image quality is great, and for the odd game that doesn't work quite right with the aspect you can squish the image.
Your best bet is to check the manual and settings of the LCD to see what display options it has, namely pixel mapping.