3/4" plywood is plenty strong enough. The issue is the brackets. They need to be very strong and, most importantly, mounted directly into studs. I would not rely on anything less - toggle bolts, expandable anchors, etc. Screws into studs.
It may be a bit tough to find the studs - I'm not sure if a regular stud finder will work through the tile. If you're not sure, drill some pilot holes in the area that will be blocked by the microwave oven. Once you find the studs, drill pilot holes for mounting brackets. I would drill just a drop larger than the screw diameter through the tiles (so that you don't crack the tiles) and then a bit smaller (so the screw can grip the studs) into the studs. Ideally you want to go an inch into the studs, so probably 2" screws - 1" into the stud, 1/2" drywall, 1/4" tile and a little extra for the brackets.
What kind of brackets? Almost anything will do, but I would generally recommend something that:
- Has support for at least 3/4 of the depth of the shelf
- Is several inches tall
- Can be screwed into the stud in at least two places per bracket
Something like this:
Depending on how your microwave oven mounts, you might want to attach the brackets upside down on top of the shelf, leaving a totally clear shelf for mounting the microwave oven underneath.
You will probably want to drill a hole (use a hole saw) through the shelf to run the power cable up to the receptacle.
The hard part will be the exhaust vent. Depending on the microwave oven and depending on your house structure (is it an outside wall? where are the studs?), you may be able to cut a hole in the wall (through the tile) for the vent. Or that may not be practical and then you would go up to the ceiling and then out (roof or to a side wall). But without an upper cabinet, any ductwork (except straight out behind the oven) will probably not look very good.