The last couple of years has seen me move my office/studio a few times. This means that the acoustic foam I use to tame reflections, also has to move. From the start I knew I didn't want to glue it to the wall as even though I own my own place; it is really destructive to pull. I don't mind re-painting, but fixing dry wall is not appealing.
I started with painters tape and that didn't work. Next was hook and loop (not sponsored and I don't want to get sued) with an adhesive. It stuck to the wall, but it came off the foam after a couple of days. After that I settled on T-Pins and those worked quite well and ended up being my initial solution. In fact I'm still using that for the foam above my listening position. The draw back however, was that they leave holes in the wall bigger than push pins. With 2-4 per panel, this left a fair number of holes to patch. If you have lighter walls it might not be too bad, but I have a dark orange and the dry wall underneath definitely stood out.
So when I moved back into the space I wanted something better. That ended up being a combination of things, but where the panel hits the wall it is simply 3M command hook tabs from the decorating hooks package. I didn't use the hooks, just the sticky stuff. This was made possible because I used adhesive on the back of the foam to attach yard sticks (cut down) from the big box store. Then the 3M command hooks stick to the yard stick. To remove I just pull the tabs and off they come. No holes or other damage! Moving to a new location is as simple as putting more tabs in the desired location and sticking them up.
Note: After some time the small tabs I've used for the textured wall have given way. For textured walls, I would use something heftier like the hook and loop tabs. Same clean application, but more strength.
Here is a quick video of me going through the process with one hand. I wish I had discovered this earlier, but hopefully this helps out someone else.