What brand name on the old MW? most microwaves have a metal plate secured to the wall that supports the microwave, with 2 screws thru the shelf above to hold the oven in place. if you have shelving paper, look under it. also check to see if a very thin piece of plywood is installed in that one cabinet to cover the screws.
Installation is best accomplished by 2 people as the unit is fairly heavy and ackward when installing the top screws.
also, if you check the installlation instructions for the new unit, you will get a better idea of how the old is installed. check back with information on the old unit for additional helop on removal