There are a few models of under counter MWs

For something under the counter, you can have a shelf and just set a microwave into it, you can also buy a trim kit for a specific microwave so it will look built-in. A trim kit would look like this:

27" Built-In Trim Kit for Sharp Microwave R651ZS, Stainless Steel, 30" · More Info

Gallery Series 2.0 Cu. Ft. Built-In Microwave, Stainless Steel · More Info

There are also under counter microwave drawers, some people prefer them because you access them from above vs the front.

Flat Panel Microwave Drawer, 24", 30" · More Info

There really are tons of options!

For the last 25 years, in 2 different homes, we had a counter microwave in a shelf/cabinet above the counter. It worked out great. For the home we're building now, the microwave will be in a shelf/cabinet below the counter since that works better with our new layout. Yes, we could do a MW drawer, but they are 5-7x the cost of a decent counter model with greater capacity and reliability.

From 1993-1999, we had a "built in" MW that was simply a counter model with a trim kit. When the MW died and we had to replace it, we had to get a different trim kit. The new trim cost as much as the MW itself. Since then, no thanks - I'll just keep it on its shelf and not worry about the trim kit.

One key thing - check the clearances required for the MW for top/back/side clearance. You can't just put them in a space if you don't provide adequate ventilation around the appliance.

We just had a drawer microwave installed in the island. this Is a great option and sounds like just what you need!

We have a garage lift. Not as expensive as a drawer or a framed micro, but not as budget friendly as a shelf.

When I remodeled my kitchen I had the microwave put in a cabinet. It looks like a regular cabinet, but it fifteen inches deep and has a clock outlet. So now I have a microwave that is convenient to use and clear counter tops. I love it.

Thanks so much - this helps. Great ideas/advice.

I have the GE Spacemaker microwave (model GE PEM10BFC). You can use this model as a built in, or install it as we did, so that it hangs from the bottom of the upper cabinets using a mounting bracket kit that you buy separately JX10PEM.

My first one lasted almost 20 years and we replaced it with the same one last year. I like that it's off the counter, so you can use the counter in front of it and under it.

following...

This was a convection OTR that the homeowners didn’t want to give up.

Beachy Morning Kitchen · More Info

Another convection MW, that could not be built in due to the heat it produces. Easily replaceable, just sitting on a shelf.

Collierville New Beginnings · More Info

A MW drawer, located below the French door oven.

Island Style · More Info

Island Style · More Info

Countertop MW built in with the appropriate matching trim kit.

Woodsy Warm Kitchen · More Info

Do microwaves need space above them or by their sides for ventilation purposes? I prefer the placement of Denita's microwave. It's high enough that I wouldn't have to bend over to use it and it allows enough countertop clearance under it to place a toaster or blender or use the countertop for prep purposes.

Yes, they need space. That is the purpose of the trim kit that builds them in, if their specs allow. See the vent grille on the built in above in the Woodsy Warm Kitchen? And the space left open around the MW in the Collierville New Beginnings kitchen? And the space below the OTR in the Beachy Morning Kitchen?

To further The Cook's Kitchen's comment, the clearances are different for every microwave. Some required 2" above, some 4". Usually a minimum of 1" on either side and 1 or 2" in back as well.

I am looking at under counter on a shelf with a trim kit - I know you need space around the MW so planning on that. I don't have the space to put it above on a shelf. Since we don't use the MW for more than heating up food I think it will be ok under counter. The price of a built-in is so much more so the shelf with a trim kit will look nice and cost less. I appreciate all the feedback. I meet with the kitchen designer tomorrow and we'll see what works. Again, thanks for all the great feedback.