Sulfur sticks. You can find them at your local hardware store in packages of 4 to 6 sticks in the pest control section of the store or you can order them online. The important thing here is using them properly to achieve the desired goal which is dead moles and this is how you do it:
In dry weather when moles are active, go out in the evening and knock down all the mounds. Go back in the morning and you'll find the fresh mounds. This is where the moles are working. Scrape off the top of the mound and feel around with your fingers until you find the tunnel. Gently scoop the dirt out of it until you have clear access to the tunnel. Light the sulfur stick and shove it as far down the tunnel as you can and then very carefully (so as to not extinguish the sulfur stick) plug the hole tightly with dirt.
Immediately move to the next mound and repeat. If you like after a couple of hours you can go back and re-open the holes to dig out any unburned part of the sulfur sticks and properly dispose of them.
You won't get the satisfaction of actually seeing the little dead mole body with this method like you would with a trap but you will do something that you can't do with traps and that is to actually kill all of your moles. This gas warfare method is so much more effective than traps it isn't even a contest but you'll hear people rail against it and I'm convinced the don't like it simply because they never get to see the dead mole. Personally I don't want to see them, I just want them gone.
Lastly I would suggest that you volunteer to perform this service for your neighbors because if you don't his moles will soon move into your newly vacated mole free lawn.