We have loads of experience with this, both live and dead ones.
Jasper eats some by the back door and leaves others for us in the house. Your cat is trying to provide for you, so you're unlikely to stop it. We've had several hundred mice, a large koi carp, a frog (live, I don't think he liked the taste), a couple of birds (he's not very good at getting these though and he doesn't like the feathers) and the other day he brought in two fat baby pigeons (carrying both at the same time)! Occaisionally he brings in a live one, but obviously with every intention of killing and eating it, he's very posessive over the live ones :-)
Delilah on the other hand is a pain in the butt. She doesn't bring in as many but they're always alive and she always brings them into the bedroom to us in the middle of the night. The other night I managed to catch one under the bin and OH ran to get something to take it out in, so I'm stood there holding the bin and another one's looking at me across the bedroom! Another time I started smelling something really bad under the bedside drawers and one had taken up residence under there! And mouse pee is dreadfully smelly and hard to get rid of!
So...from great experience, one or two pieces of advice:
1) your not going to stop it so be prepared! Keep the floor as clear as possible so live ones don't have places to hide, and
2) have some sturdy gloves ready (bites hurt and can carry diseases)
3)Keep the cats flea protected and check regularly for ticks, fleas and injuries around the face (I wipe them with tea tree wipes every other day and give them a johnsons flea tablet if I see any other signs). If the dead ones are on carpets it's worth spraying with flea spray just to be sure.
4) If it turns out you do have an infestation, please DONT PUT POISON DOWN as one person suggested, they don't always die straight away after they've ingested the poison, but they do get slower and easier to catch meaning your poor baby has now eaten a poisoned mouse. I had to have words with the neighbours over this very same thing. Instead use traps. Be careful clearing out nests, mouse and rat pee carries diseases and the little sods bite as well.
5) Deworm regularly, most wild animals carry parasites that your cat will pick up (and worms are totally gross so prevention is better than cure).
6) If you have to clear up half eaten ones, again be careful about disease control and what disinfectant etc you use (not all are pet safe).
Sorry if any of this is stuff you already know, I thought it best to cover everything I could think of, hope it helps :-)