using a French cleat would work. They used to make some very small ones, but can not find them. Also could use a picture frame keyhole type bracket.

Here is a link that might be useful: french cleat

The panels need to be flush against the wall, so I think the cleat won't work. A keyhole bracket might...looking at one from Rockler, some reviewers have recessed it into the surface, so it can be flush.

Would you put the bracket on the mounting surface or the hanging panel?

Here is a link that might be useful: keyhole bracket

This post was edited by weedyacres on Mon, Jun 23, 14 at 16:02

I would recess the bracket on the hanging panel, and put a screw into the wall.

I've used the Rockler brackets and they do work as advertised. Obviously the screw they attach to needs to be positioned very carefully you may need a couple of "adjustments" to get a really tight fit.

I just read some more on the Rockler brackets and they come with 5/8" mounting screws. If the bracket is recessed on the hanging panel, that only leaves 3/8" thickness for the screws. Do 3/8" screws exist? If so, is that really advisable? Would I be better off putting the brackets on the wall?

If you have some wood scraps I would test the fasteners on that.

Hot melt glue a 1x cleat to the location of the bottom of the panel. Space some quarter-sized gobs of construction adhesive or silicone on the back of the panel in 16" intervals or so and set the panel on the cleat. Place two load lock jacks in front from floor to ceiling and drive shims between them and the panel to clamp the panel to the wall. Pull the jacks and cleat the next day. If the hot melt adhesive sticks to the laminate scrape it off with a credit card and clean residual with acetone or denatured alcohol.

Here is a link that might be useful: Load locks

Trebuchet--then how does weedyacres get the panel off in the future?