Actually, it is kind of inaccupuncture, since there really isn't any attempt to poke at particular neurological points in the body. It's actually sort of random and all encompassing. The most effective method is to fall over on top of a Cholla. You can choose either the teddy bear or the staghorn, depending on your preference.
My option this time was the staghorn. This is a fine example, not the one I chose.
To give you a better idea, here's a section of a staghorn.
I heard a suggestion on how to remove the smaller spines that you can't quite get with a pair of pliers that you of course have in your toolkit. Spread wood glue over the area (something non-toxic like Elmer's), let it dry, then peel it off, You'll also probably lose all the hair under the glue also.
Let's be clear about riding in the desert. Everything out there wants you dead. That includes rattlesnakes, gila monsters, spiders, scorpions, the weather, cacti, trees, bushes, you name it. Humans are intruders and nature doesn't like you in the desert.