The color of a piece of raw meat depends mostly on the type of muscle tissue it was cut from and not on the diet of the animal. Keep in mind that some animals, like chickens, provide both dark and white cuts of meat. Darker meats, such steak, chicken legs or duck breast, are made up of cells that store a lot of oxygen for sustained activity, like for a cow "out standing in her field", a chicken running around pecking at seeds, or a duck flying for a long migration. White meats, such as chicken breast, are cut from muscles that didn't get used as frequently by the animal (chickens don't fly very much) and thus those muscles didn't need a constant supply of oxygen. Once cooked, the taste of a piece of meat depends a lot on how much exercise the animal's muscle tissue experienced. For example, veal cooks faster and tastes tenderer from other beef cuts because veal is cut from calves that were not exercised.

This is a very good question, and the answer actually has to do with the composition of the different meats at a molecular level! As you may know, the world around us is made up of different types of atoms, and these atoms assemble together in different ways to form molecules. Interestingly, the exact way that the atoms are arranged in the different molecules causes them to appear to have different colors (and smells, tastes, etc...). There is a certain molecule in meat which gives its red color, that molecule is called myoglobin. Since the molecule is found in all types of meat it is the number myoglobin molecules that differ between the meats, giving them their color variation. The white meat of chicken has under 0.05% myolglobin; pork has 0.1-0.3% myoglobin; and beef has 1.5-2.0% myoglobin. So you can see that as the amount of myoglobin in a meat gets larger the color of the meat appears redder!

The diet of an animal does not make a big contribution to the animals myoglobin content; however, the fitness level of the animal may play an important role! For example an "in shape" cow may have a higher myoglobin content (making it redder) than an out of shape cow. This is because the myoglobin molecules in meat are responsible for transporting oxygen into the meat, and the more "in shape" an animal is the more myoglobin molecules it needs.

Great Question!
What we call meat is actually muscle. Muscles have a few different things in them, such as blood vessels and the actual fibers that contract when the animal moves. The type and percentage of these things makes the muscles certain colors.

For example, you probably already know that poultry (like turkeys and chickens) have dark meat and white meat. Chickens walk around most of the time. This is an aerobic activity. "Aerobic" means that the chicken breathes in enough oxygen to keep up with what they're burning. So the muscle fibers in their legs are a type that uses a constant supply of oxygen, and doesn't tire out for a long time. The muscle cells have tiny organelles in them called mitochondria that use lots of oxygen, but turn food energy into the energy needed to work the muscles.They also make the muscle darker. There are also lots of blood vessels to carry oxygen into the muscles, making them darker too.

Chickens only fly occasionally for short distances to avoid danger.This is an anaerobic activity. "Anaerobic" means they use oxygen faster than they can breathe it in, like you do when you run as fast as you can. So the muscles that power their wings are a type that contracts quickly and can work without being re-supplied with oxygen, but they run out of energy very fast.

I'd guess that there are similar reasons for the different muscle colors in the different animals. Why do you think there's no "white meat" on a cow?

Do you think that geese, which fly very long distances to migrate, have white meat?

For some animals, yes, but not for the three you named.

There are two different kinds of muscle cells, one which is adapted for sudden bursts of strength, but which tires quickly, and another that is less strong but can continue to be active for much longer. The former type is white, and the latter is red. Most muscle is a mix of both, but how much of each type determines the color of the muscle. Chickens, which no longer use their wings, do not need great endurance in their flight muscles, so those muscles have lost their enduring fibers and turned white (unlike their legs, which are dark). Cows are built to roam and wander great distances - remember, bison are basically wild cows. Their muscles are made for endurance, and have more of the red, long-endurance fibers. Pigs are somewhere in-between.

The main classification of meat is either red or white. Pig meat (pork) is actually red when raw and then turns white when cooked so it's not really a different like pink.

In poultry (chickens, ducks, turkeys, etc.), both red and white meat are possible. The more muscular parts that are for doing things for a long time (endurance activities such as flying), use more oxygen; this requires a large amount of myoglobin. This gives the meat a red color. Muscles that are used for short bursts of power do not require as much oxygen (hence less myoglobin) so they have a white color.