It's very difficult to photograph so I never did the planned photo essay.
In short you will need one 18x24" sheet each of 1" and 1/2" adhesive-backed acoustic foam sheet, which you can get from Amazon.
Amazon.com: Sonic Barrier 1" Acoustic Foam w/PSA 18" x 24": Electronics
Amazon.com: Sonic Barrier 1/2" Acoustic Foam w/PSA 18" x 24": Electronics
The thicker foam is trimmed and placed on the outside wall behind the microwave cabinet. This area is accessible by removing the microwave. That controls most of the noise that emanates from the vent above the microwave.
The thinner foam is pieced in in several key places. Access is poor but can be achieved through the drawer under the kitchen counter, the kickspace air return, and the two forward-facing heat outlets (you have to remove the registers but not the ducts. I avoided placing any foam on the furnace itself since, even though it is flame retardant, I was concerned about the effect of heat. The locations were:
1) inside the lip of the kickspace return, that is, the part that hangs down to within 2" or so of the floor. I measured, and Atwood's return air specification is still met, because of the vent above the microwave.
2) the casework aft of the drawer. Easily accessible by removing the drawer. Helps a little with reflected noise.
3) the casework behind (starboard of) the drawer, again easily accessible, small gain but not much work.
4) The inside of the casework where the two heating ducts come out. This one is difficult to do but makes a big difference. You have to cut holes for the ducts.