I’m assuming from the layout that the wall it a free standing wall with no way to run the wires behind crown molding or baseboard trim.
You then have three options; 1) rout it above the ceiling in the attic 2) under the carpet 3) in the crawl space.
I’d go with the crawlspace if you have access. It would be fairly painless job, just be careful. You’ll need to first see what’s under your floor joists so don’t go poking holes in anything that ain’t supposed to have holes. Take a few barring measurements so you know where that back wall is while you’re under the floor. You will need to drill up through the sub floor to the wall locations under the receiver location and the back wall where you want to mount you speakers.
Go back to the living area and drive a small (ten penny) nail through the floor as close to the wall as you can get in line with where you want to mount the speakers and rout the wire from your AVR. The nail will be your reference when you go back to the crawl space to drill your holes. Measure over from the nail 2” and drill a ½” hole up through the floor. Don’t go any further than you need to go to get through the wall bottom plate. Shouldn’t be more than 2 ¼”. Go real slow and careful because your electrician may have stapled the romex to the bottom plate. You will know when the bit is about to bust through as it will start to bind at the very end when you drill slowly. When you’re almost through, poke a screwdriver the rest of the way just in case there is romex there. Don’t get crazy, just hard enough to bust through the last bit of wood.
After the holes are drilled, you need to cut in some ‘old work’ J-boxes. Get the ones that have a butterfly expander on the back. Just ask the sales dude at the hardware store and tell him you are retrofitting J-boxes in sheet rock. He’ll know what you need. When you cut them in, cut them in only as big as they need to be. They should fit snugly. After you get the holes cut, go back to the crawlspace and run a fish tape up the wall, keeping it up against the wall board. Shove it up there far enough to reach the J-box location. When you go back upstairs, grab a metal hanger and snag the fish tape through the hole.
From there it’s a slam dunk piece of cake and you will save yourself about $300 in labor. - At least that’s what my estimate would be.
If you have to route them through the attic, the same technique applies. Just make damn sure to not step on, kneel on or sit on the sheet rock. Stay on the tresses!
If you have any more questions, just holler.
