If you have something flat that you don't want to put a hole in, a key-style picture hanger is a good option (does anyone know what this is actually called?). It involves a small length of board cut in half at a 45 degree beveled angle. You stick one piece of this board to the back of your flat thing with the long end on the outside and facing down, and the other half, you attach to the wall or surface on which you would like to hang said thing. It looks something like this:
The beveled edge acts like a clip, holding the thing in place. You can pick whatever fasteners suit you, and it's easy to get the whole thing level by leveling the board first.
If you don't have a tool that makes nice beveled edges, you can use a bit of tongue and groove flooring in the same way. Just cut it in half (or thirds, as I did), flip the halves over, and you have a nice key system. The two pieces should lock together nicely. Once you procure a saw and some flooring, the rest is pretty simple.
Step 1: Safety. Eye protection, hearing protection, gloves, dust mask.
Step 2: Cut in half or in thirds (as shown).
Step 3: Remove waste
Step 4: Flip halves over
Step 5: Test fit
It seems to me like the tongue side should hang on the wall, and the groove side should be attached to the thing, but I'm pretty sure it would work the other way, too.
Apologies for taking liberties with perspective. It's late.
The tongue and groove design makes it easier to use those removable double-sided tape things because the tabs don't get in the way of the clip.
The thing will end up being set off from the wall just a bit.
I hope that somebody out there finds this useful!