Do you mean fantasy stories in which there are animals capable of speaking english, or stories in which the point of view is an animal? (and the animals then talk to each other). The latter is actually my sub-genre of choice--I grew up loving animal stories, and I'm just starting to get into writing them now. I think a well-done animal pov story can be wonderful--sure, there's the cheesy dregs out there too, but that's true of every kind of story. But the neat thing about animal stories is you can see the world from an alien perspective.
Watership Down is, of course, a classic, and is the poster child for animal stories. And although yes, the author had to give them sentience and language (an example of an animal story which doesn't do this though is Raptor Red, which was a very different and fun read), the story is still very much told from a rabbits perspective. They view things in ways a human wouldn't, and have motivations and concerns that are centered in a rabbits world. This gives the reader something exciting from a fun and interesting perspective, and at the same time, gives them something to think about.
Firebringer is a story of a similar nature only about deer; Whalesong, and it's two sequels, was a lovely story about humpback whales (unfortunately, I don't think these are in print anymore). Silverwing and it's two sequels are a totally awesome and fun story about bats. The White Bone, which I'm reading right now, is a sad and beautiful story about elephants--which deals with the real-life issues of elephant slaughter in a way that wouldn't be as powerful if it were told by a human observer. All of these (and more) are serious stories, and none of them are cheesy. You come to care about what happens to the characters just as much as if they were human, even though you are always aware that they are not. That's a sign of good storytelling.
And dammit, animal stories are just fun. ^_^