I can get hormone-free chicken legs for 1.99/lb at the local Whole Foods. My big dog eats about a pound per day. The small dog eats a half pound. Since high-end dog foods cost $2/can, and I was feeding my big dog 2 cans a day and my small dog 1 can per day, it has basically cut my costs in half.
I also get them chicken gizzards and hearts and beef liver since some organ meat is necessary.
If it's on sale I'll buy bone-in pork chops, some beef stew meat, or some talapia, all of which are also relatively cheap (especially if they get close to expiring).
I've dedicated the bottom drawer of my freezer to storing the stuff, and the bottom drawer of my fridge for thawing it out. I just open the packages in the backyard and throw the meat on the ground. That way I don't contaminate my kitchen.
I also give them a couple raw carrots after they are done eating the meat to freshen their breath.
If you try it, I'd start with raw chicken since their rubbery bones are the easiest for dogs to digest. Some dogs have trouble with big hunks of beef or poke bone. They will swallow too big of a piece and it will damage the intestines on the way out. But if you've ever given your dogs bones as treats, you should know if your dogs can handle it or not. My big dog will occasionally swallow more bone than she should, but then just throws it up a few hours later. Also, never cook any of the meat. It makes the bones brittle. The brittle bones will splinter and cause damage. Raw bones don't do this.
Using this diet, they essentially digest everything but the excess bones. As I said, the shit turns to white powder in a few days and doesn't stink at all.
If you want to stick with regular dog food, any grain-free variety is fine, and will drastically reduce the size and nastiness of the shit, since dogs can't digest grains and just pass them out the other end.