Examine the radiator cover to locate screws or other fasteners holding it together. Check the radiator register for screws and fasteners on the outside; most of these are on the inside of the radiator cover, however.
Make sure the radiator is off and cool enough to work around. Match the screwdriver tip to the screws -- if the screw has a single slot, use a flat head screwdriver; if the screw has two intersecting slots, like a plus sign, use a Phillips head screwdriver.
Unscrew any fasteners holding the radiator cover to the wall behind the radiator and any holding the radiator cover to the floor. If the register has exterior screws holding it to the cover, simply unscrew them, remove the register and proceed to Step 5.
Carefully remove the radiator cover and lay it face down on the floor or other flat surface. Unscrew the register from the front of the radiator cover and remove it.
Measure the radiator register using the measuring tape and write down the register's dimensions on a notepad. You will need the measurements to purchase a replacement register that is the correct size.
Place the old register aside. Put the radiator cover face down for an inside-mount, and position the new register face down where the old register was. Screw it into place. For an exterior-mount, screw the new register into place on the outside of the cover.
Reattach the radiator cover to the unit, as well as to the wall and floor if it was previously attached at these points.