Experts agree that protein, usually in the form of meat, should equal or exceed vegetables in a dog's diet -- they just don't agree on how far this should go. They also appear to disagree on the definition of vegetables; some include grains as vegetables and some separate the two.
Dogs are basically carnivores that have become omnivores through living with people. They can eat most of the things we eat, but they need more protein than we do. People can do well on a vegetarian diet and dogs can survive, but not thrive, on one. Experts recommend that anything from 1/3 to 3/4 of a dog's diet be protein, in the form of meat or non-meat protein, such as eggs, cottage cheese or cooked dried beans. Poultry (chicken, turkey, duck) and beef are the most common meats to feed dogs, but they'll also eat things that most people don't care for, from rabbit to possum. Organ meats, such as liver, heart, gizzards and tripe, are also acceptable and very good for dogs in moderation. Here's a good chance to use up that freezer-burned roast or the venison your hunter friend gave you that you don't really fancy.
Green and yellow vegetables and root crops other than white potatoes can make up anything from 1/3 to 1/4 of a dog's diet. These are best cooked to make them more digestible. A dog's digestive tract is shorter than a human's, so they have less time to extract nutrients from raw vegetation -- salads have no place on their menu. Pumpkin is a superfood for dogs, and while dogs aren't usually big fans of leafy greens, you can sneak these into a stew and they'll eat them along with the rest. You can also include some fruit -- offer apple slices (no seeds) as a treat. Vegetables supply vitamins, minerals and fiber.
Grains can make up 1/3 to 1/4 of a dog's diet, depending on his activity level and his need for quick energy, which is what carbohydrates supply. Whole grains, including rolled oats, barley and more exotic types such as quinoa, are good as long as they're cooked thoroughly, but one of the best and most readily available grains is rice. Brown rice is best for everyday meals, since it supplies some B vitamins and a bit of fiber from the seed coat, but white rice cooked in chicken broth makes a bland diet for a sick dog or one who's just off his feed.
The Other Ratio
There's another important ratio to be considered in a dog's diet -- that of calcium to phosphorus. Ideally, this should be 1:1. If this ratio gets too far out of whack, your dog's body will take calcium from his bones to make up what's not in his diet, and this can lead to osteoporosis and other skeletal problems. Meat, especially organ meat, is high in phosphorus and low in calcium. If you feed a home-cooked diet but don't give your dogs raw bones to gnaw, you need to add a calcium supplement in the form of powdered eggshell or calcium carbonate (indigestion tablets). Your vet can advise you best on how much your dog should get, but 900mg per pound of food is generally considered adequate.