I installed soffit vents in the cabinet where our
microwave sits, but added ducts to direct the fan-driven exhaust out of the cabinet. I'm not sure is you'd get enough circulation if you were relying on plain-old passive convection to move air through the closet.
What we did was to make new doors with punched metal panel inserts (available at Lowes & Home Depot) in them. They are easy to make and allow tons of circulation.