Classical "Zombies" - undead being which feed on flesh, and can only be killed by destroying the brain - cannot exist.
Basic biology forbids this.
When you die the normal processes which power your body end. Your heart stomps pumping blood. You stop breathing. Without blood distributing energy and oxygen to your muscles, they can't function. If they can't function, you can't move.
It doesn't matter if a Zombie eats brains, or anything else. If that food is not broken down, and the energy not transported to the muscles then you can't operate the body. The whole system needs to be functioning, not just convenient bits and pieces.
Within minutes of the brain being deprived of oxygen our neural pathways begin to deteriorate. There is a very real risk that if you reanimate someone after they were deprived of oxygen for a long time they may suffer from severe and completely unforeseen brain injuries.
Maybe they will have lost their memories, maybe they will be "alive", but never wake up from the coma they're in, etc.
The brain also "runs the whole show" as far as operating the body goes. There is no way that a dead body could rise after months of undergoing decomposition. Even after days of decomposition. There would be nothing left to "think" about eating or anything else.
A virus which makes people go crazy and attack others is not that far fetched. Watch the movie "28 days later" for a pretty chilling view of what could easily happen if the right animal activist breaks into the wrong lab.
These are not zombies, however, in that they will die of the same things we do: malnutrition, injury, poison, disease, viruses and bacteria - even the common cold could do them in.
A bullet to the chest would be just as effective as one to the brain (in the grand scheme of things).
They would still need to go to the bathroom, except they wouldn't remember how to take their trousers off, or how to wipe their asses. How long would it take for some severe and utterly nasty infections to set in?
If you're dead set on setting the undead loose on the world, then nanotechnology might level the playing field for you.
Nanobots wielding the bodies of dead humans and animals in order to fight their creators (us), could sound somewhat plausible in a sci-fi setting. I say this because you can claim that rather than having the body move under the power of its own muscles, nanobots would actually be facilitating the movement, not living cells.
All you'd have to do is explain why those nanobots did not simply choke the life from every human in the first place, or why they don't build robots to murder us all in our sleep, rather than shambling about in rotting corpses.