[http://www.mydigitallife.info/2007/10/04/how-to-reset-iphone-and-ipod-touch-hard-reset-or-reboot/|From here] "The iPhone and iPod Touch may become completely unresponsive with LCD screen fading away to blank or black. And the gadget device may also does not accept nor respond to any screen taps, and ignoring or not responding to any external buttons pressed. In this total lockout situation, a possible quick solution is to reset the iPhone or iPod."
So my suggestion before replacing anything would be to reset the iPod. You accomplish this by press and hold the Sleep/Wake button and the Home button at the same time until you see the Apple logo show up on the screen. Keep holding it, sometimes it takes a little longer. This of course will not work if anything happened to the iPod prior to the LCD not functioning. It also does not mean that the LCD will not have to be replaced as Scott Millard suggested, but give this a try first. Good Luck