We have a digital thermostat that controls our air conditioner, which until recently was on the wall in our living room. Although we don't have a definitive floor plan for the living room yet, we decided the thermostat would be better off in the hallway, which would open up the wall in the living room for taller pieces of furniture, which would otherwise block the thermostat.
I wasn't sure what I'd find when I removed the thermostat from the wall, but was relieved to see that there was only a small hole in the paneling for the wires to pass through. I loosened the wires from the thermostat then drilled a new hole in the hallway wall.
Getting the wires from the living room through the small hole in the hallway would have been a challenge if it weren't for the convenient placement of the AC return. I opened up the return cover and (after thoroughly checking for bugs with a flashlight) squeezed myself inside the wall, where I pulled the wires out of the living room wall and fed them through the new hole in the hallway.
I rewired and mounted the thermostat in the hallway and patched the hole in the living room with a little wood putty.
Although our primary motivations for moving the thermostat were for looks/furniture placement, there's also a practical benefit of the new location. The best placement for a thermostat is often near an AC return. This is because the air at the return will be more representative of the air in the home, as the air is pulled in from around the home, and should result in a more consistent temperature. It also makes sense in our house since we only have one AC return, and it's in the center of the house.