I don't know what I thought was so difficult/mystical about the whole process, but I'm kicking myself for not doing this before. I washed the yeast from an Irish Red I brewed a few weeks ago, and then re-used two jars right away for a modified version of the Cream of Three Crops yesterday. Today, fermentation is going strong!
Granted, the yeast I washed was the Safale-05, but it was a nice experiment and encourages me to give it a shot with different yeasts.
I've got 16 mason jars just waiting for more yeast