Acoustic Foam Panels - Blu Tack ???

If you have sheet rock walls, medium length common pins will work fine for smaller panels, say up to 2' x 2'. Be sure the head of the pin is secure (many plastic tips will come off rather easily), and that you try to distribute the weight evenly using multiple pins. For bigger panels and/or items like rigid diffusion panels, I use quiltmaking "T" pins. All leave minimal marks, are easily repositioned, and hold a surprising amount of weight.

-Richard

I recently bought some new acoustic foam panels here and found that command strips do a good job of holding them up without causing damage to my walls. Before that, I took a load of paint out and had to fill it all in. It took me ages to find the right shade of white to cover it up.

I'm guessing that the OP has found a solution in the last 14 years, but here's what I recently bought to deal with this same issue -  https://xfasten.com/collections/best-sellers/products/xfasten-acrylic-mounting-tape-removable-1-inch-x-450-inch

It seems to be more than strong enough and peels off without leaving any residue.  It's similar to the little double sided removable glue dots, but stronger.

Thanks big_greg for the tip on the XFasten tape. I'm going to give it a try.