Would you help me gather ideas to help me hang flat wood panels on the wall?

Use paneling glue on the back and on the wall, then put up on the wall. Once in place tack it down with finishing nails.

Are they permanent or temporary?

Temporary ill upload photos soon!

I’m thinking furrying strips across the wall that you could remove as needed, but screw or nail the panels to the strip to protect the walls.

Here’s one http://tinypic.com/r/i1wk94/5

Those look heavy. Make sure you hit the wall studs with the screws or whatever you attach the furring strips with.

Similar to @Adirondackwannabe‘s idea you only need a “hanger” board horizontally mounted.
Given the image it looks like you can rest the frame of the picture on the hanger board.
If it needs to be secured you can sink a screw or nail down from the top into the hanger.

Again I agree with @Adirondackwannabe, find the studs ! Or at least use serious anchors or toggle bolts.

I also have a much lighter cupboard door that is flat (used to paint on). I’m not sure how to make it hang in a gallery…

You want a french cleat. There are many tutorials on how to make one.

@adirnobe the one in the photograph is actually heavy! They are 26×26 inches and weigh about 6–8 lbs
@auggie great fund thanks so much!

I have been using french cleats forever but never knew what they were called. You can hang just about anything with them.

@augustlan “French Cleat” best idea !
I’ve seen kitchen cabinets hung from french cleats, that’s the ticket. Very strong and unless you’re in earthquake country you wouldn’t have to further secure it.

Wouldn’t the person whose gallery it is answer this question better than we could? Don’t they have a preferred hanging system?
French cleat is genius, but it requires the gallery to install the half that the top half hangs from. ... confused here.

You’d want to mount the other half of a french cleat on the painting, too.
If this is a temporary installation and you don’t want to use a cleat after the exhibit it might be unnecessary damage to the painting frame for a temp application.

@susanc has a good point, the gallery has experience hanging in their space.
It seems they could lend some direction why not?

For home decor or any fairly permanent installation a cleat of some kind is something to consider.
You can center the cleat on the painting and sink screws into the wall where there are studs in the wall. If the cleat is long it enough, it will overlap at least two studs.

I hung a 4’x4’ canvas (that’s on a frame like that in the picture) on two big pan-head screws, screwed into substantial anchors, meticulously levelled. I left the screws about ¼” out from the wall and the screw-heads bite slightly into the frame when it’s hanging. There is no modification done to the painting frame.
It’s very easy to take that painting down, for cleaning or wall maintenance or lending to the Tate (ha! made that last part up), and even better it’s easy to put it back up.
It’s been up there four years now.