In theory, yes. Animal mobs will spawn anywhere there is grass and enough light.
However grass will only grow on blocks that are adjacent to other grass blocks and no more than two blocks above or below. If no grass is present at the start of a new world, as I believe is the case in snow world, grass will never grow in that world.fseag
I think that's the problem. Not completely sure, but I think that neutral mobs can only spawn on GRASS, and the problem getting grass underground is that it requires SUNLIGHT. It would probably work if you'd make a glass ceiling in that area. That would make it possible for sunlight to gain access to the dirt, growing grass on it and spawning mobs. If somebody is more experienced in MineCraft than me and knows more exactly of animal spawn requierments, feel free to correct me.
Dirt tiles don't need sunlight to grow grass, all you have to do is have a grass block somewhere on the surface and make a dirt-staircase from that one down to your underground mob spawn. If you places torches all over that staircase, grass will (eventually) jump down to each block on the staircase, and reach your underground area. That said, it will take a hell of a long time for that to happen, though it depends on your luck. I once set up a grassy area just 5 blocks below the surface and it took a few ingame days for the grass to reach it.
Also, I'm pretty sure snow worlds still grow grass, its just buried under the snow. Otherwise there would be no friendly mobs in snow worlds.
Check out the image in this article for more information on grass and how to grow it. http://www.minecraftwiki.net/wiki/Grass
Grass does NOT require sunlight; any bright enough light allows it to spread. I can post images if you really don't believe those of us that agree, but I have grass underground. As to the spawning, it only takes lit grass to spawn animals. I accidentally created an animal farm on a snow world by having a dirt floor in my house (to which grass spread). To fix the misconception about snow worlds: they have grass, it is just under the snow. You can clear the snow with your hands or a shovel or a torch to see it.
To start your farm: build a roof over a dirt floor (at least three blocks high, more is better), make sure it is well lit with torches every 5 or so blocks. It is best to build the roof over a spot that already has grass, but if not you can invite grass in with an open doorway and a torch outside. Grass will spread down (or up) stairs as steep as you can jump.
Sometimes I think I have gone mad, other times I just wish I had.