Of course, the internet offers a ton of suggestions. Hold your breath. Put your tongue against the roof of your mouth and breath through your mouth. Submerge it in water for a few hours before cutting. What a hassle. Finally, though, I seem to have found something that works pretty well and is dead simple.
Oddly enough, its from Gordon Ramsey. He's not the only one to suggest it, mind you. Cutting down through the root seems to be the trick. I'd always been a doofus and hacked the root right off, releasing the tear gas contained inside. Silly me.