Read the microwave oven's instructions to determine whether or not the manufacturer provided a template for cutting the vent hole. If a template exists, tape it to the exterior wall where you plan to install the microwave oven. If no template exists, measure the location of the vent, and transfer the perimeter of that measurement to the exterior wall with a pencil.
Put on safety glasses, and turn on a hand-held rotary tool with its cutting blade installed. Press the cutting blade against the outline for the wall hole. Move the rotary tool slowly and steadily around the outline to cut the wall material. Stop cutting when you reach the place at which you started cutting. Turn off the rotary tool. Pull out the portion of the wall you cut, and discard it.
Insert a 5- to 6-inch long drill bit into a power drill. Use a masonry bit if the exterior wall surface is brick or stone. Use the power drill to create a hole at each corner of the wall area you cut out until you penetrate the wall to outside the house.
Walk outside the house, and locate the four holes you drilled. Use the holes as a guide to cut a circular vent hole on the exterior side of the wall. If the exterior is wood or siding, then use the rotary hand tool or an angle grinder. If the material is brick or masonry, use a hammer drill with a masonry drill bit to make a series of holes, forming the perimeter of the vent hole. Remove material from the hole, using a hammer and chisel if necessary.