Here's what's often done for shipping delicate telescope mirrors cross-country. It might work for LCD monitors, too:
Go to Home Depot or Lowe's and buy a 4x8 sheet of 2" thick styrofoam insulation. Add 2 inches to the length and width of the monitor and cut at least 3 pieces of styrofoam to those dimensions. In one of those pieces, cut a hole just large enough so the monitor fits snugly. Then make a sandwich using the 2 uncut pieces on the outsides. Note: if your monitor is more than 2" thick, you'll need to cut additional center pieces. Now you're ready to double-box and ship.
The key idea is to have at least 2 inches of styrofoam "padding" surrounding the monitor in every direction.