From LifeHacker today:

Using a knife, cut a cone out of the bottom of the onion (where the roots come out). The diameter of this cone should be about a third of the diameter of the onion, and about 1/3 deep. Take this piece and throw it away (don’t put it down the disposal!). This piece contains the part/gland that makes baby Jesus and everyone else in the room cry when you’re chopping it up. Once you’ve gotten that piece out, chop off the top, peel, and slice the onion.

Been doing it for years, and it works like a charm. You know your cone is too small if it doesn’t work, because you’ve cut into that teargas grenade.

I always cut my onions this way anyway...and have never really had an issue with tearing up when cutting them. This must be why...