After giving the hot water method and its iteration a real college try and failing, I tried baking soda and vinegar.
I made a liquid using about 2 tablespoons of baking soda to 1 cup of water, poured it over my scalp, let it sit while I showered, then rinsed it off with lots of water.
I found doing this step alone makes for brittle hair like nothing I've ever experienced, so I tried a vinegar rinse. I mixed a half cup of apple cider vinegar with a half cup of water, poured it thoroughly through the hair, and rinsed with warm water again.
Why apple cider vinegar as opposed to other types of vinegar? I can't tell if there is a difference. I found the suggestion on a blog, and it works pretty well, so I'm sticking to it.
The result is....ta-da....clean hair! It doesn't feel quite as squeaky clean as shampooed hair, but it's as a close second as I've found so far.