Moe, that does suck.
I have a protective cage around mine, sort of... let me explain.
My old monitor, a 19" tube, sat on my desk, by a window. Well, see, the pet cat got used to laying on the nice warm monitor by the window and looking out, especially in the winter (even though it doesn't get that cold around here)
One day, my old Viewsonic 19" bit the dust. Time for a wide screen flat panel. Now I knew that cat would have to get used to the idea that she couldn't lay up there anymore. I showed her the top, behind it. Carefully coaxed her to jump up and down out of the window sill. She actually got used to the idea. Then she forgot. Well now I have a custom made shelf I made, stained and varnished to match the desk, with a padded top, straddling my LCD monitor, and securely fixed to the desk so it won't move. SPOILED CAT.