It doesn't make sense that this "virus" is only affecting humans? Why isn't it affecting every living thing when they die? Animals have the same sicknesses as us, a little varied I might admit but still. Is it that they have the cure? Or is it because their brains aren't like ours and due to the fact we have a different type of brain? Please specify whether you're talking about the TV series, books, or comics please.
The Walking Dead borrows heavily from the (mass outbreak) zombie genre. In very few other movies or stories have animals ever been able to become zombified. Though they may come up with their own particular explanation for why this is so, I contend that they have it so only because no other stories have ever done it.
And the reasons for that are interesting. Zombies that are truly undead (that is, there is no biological or scientific explanation for them) have had their original souls depart the body and something far more sinister animates them, though it has not returned life to the corpse. I have always gotten the impression that they were hinting that this happens because something defiled the body, either while it was still alive, or after it had died. (And being gored by another zombie likely defiles your body too, thus the infectious nature of it.) Animals cannot be defiled, as their bodies aren't sacred in the way that human bodies are.
But that doesn't satisfy the modern audience, I do not think. Hence Walking Dead's use of a pathogen detectible with medical imaging (as seen in the season one finale). If that explains zombification, then it's not much of a stretch to imagine that the pathogen has no zoonotic tendencies (in layman's speech, it can't be transmitted between species). Of course, this just raises other questions and mysteries, if that happens to be the case.
Note: I don't read comics, and this is based only off of the AMC television show. Several storyline differences have arisen so far, and this could be one of them.
It could be as simple as how animals carry diseases but aren't affected (Swine/bird flu, malaria, etc.), but once a human is infected they get the symptoms of the disease. Since the DNA and number of chromosomes varies for every species, it can change the way it affects all of them. Humans might have a certain chromosome that can be affected by the disease, that other species don't have.