If your getting whole animals brought in your lucky. Our cat tends to leave us nice morsels, some unidentifiable, usually places so I step on them the minute I go out the back door.
I don't think you can stop it. I've had cats, well my mum had them since before I came along and if a cat goes out it will do what nature intends. Our cat has a bell but usually the fact he is too damn lazy helps out the local wildlife. He still manages it though, perhaps the animals climb into his mouth, must be the only way I can imagine him catching anything. I got told once that it was the cat showing its appreciation by bringing back a "treat" for the owner.
I wonder when the cat will realise no one else in the house would find a mouse head or squidgy bit appealing?