I want to install a microwave over our stove, which happens to be on an island. I was told you cannot do this. I am not going to be using a standard cabinet. I will be building the cabinet entirely out of 3/4 material (to include the back), everything will be srcewed and glued together. I will also run bolts though the top of the cabinet to suspend it from an attic joist. It is going to be a well built unit and I think up to the job. :)

There's a "microhood" and a "microwave", with the first being both a microwave and also a vent hood/range hood. The latter is nothing but a microwave and has no capabilty to vent heat / cooking odors, etc from the stove. I think you meant to say "microhood" ? And if that is the case, then I see not reason why you can't do that, just check with the city, county to make sure it's up to code and I am sure they will tell you just as any contractor will also probably tell you, that you should/must have a way to vent the stove. By the way, it's my understanding that some "microhoods" (if not all) have the capability to vent heat/cooking odors through the top of the hood into a vent leading somewhere outside your home.