My previous experience in cutting onions involved myths about sticking a piece of bread in your mouth, whether to cut the top or the crown first, breathe only through your mouth, and I even resorted to wearing swimming goggles!
I honestly think mine is simpler. Sunshine goes into detail on cutting technique, whereas I simply keep my onions in the fridge.
Assuming you are chopping about a dozen onions for a barbecue or dinner party, have an empty bowl on hand and select your first victim from the fridge. While it remains cold, you can peel, cut, slice, dice however you please. Now put the result in the bowl and return it to the fridge when you select your next onion. This onion bowl is only going to be sitting in the fridge for the time it takes to cut all of the onions, so a lid is optional.
Once you have your desired amount of cut onions in the bowl, you're ready to toss them into your recipe. This is when tears may now arise because of the heat. Thank heaven for the extractor fan in the cook-top!
When done, you will still need to wash your hands thoroughly to get all the onion traces off and, of course, you did wash your hands before starting as part of your kitchen hygiene routine, didn't you?