I used to use Dr. Bronner's to wash the dishes.
We had 2 gallons Dr. B's bartered to us from people who owned a health food store. We brought the soap with us when we moved, and for about a year after we moved we were B-R-O-K-E. Too broke to even buy dish soap. But we got food stamps, so I could get vinegar.
I filled something like
this with straight Dr. Bronner's, washed in very hot water, and then rinsed in warm 50%-50% vinegar water.
It was okay if I towel dried everything. Air drying allowed a film to form. Once we didn't live in poverty anymore, we bought a dishwasher. I use an OxyClean/lemon Kool Aid blend in the dishwasher, and, as a consequence, use far less dishsoap, but I like keeping Dawn around for its grease cutting properties. However, if I ever get around to mothering my own vinegar, I'd be willing to let the Dawn go.
My neighbor, who isn't that old but grew up in a house without indoor plumbing (in the 40s-50s) told me that, in order to wash dishes, they used two basins and first gave everything a very complete scraping. It went dirty stack of dishes: wash basin : rinse basin: clean stack of dishes. They used a crocheted dish cloth that they rubbed right on a bar of something like Fels Naptha, use the cloth to scrub the dish, rinsed, and towel dried right away. Really greasy stuff they'd bleach.
Not terribly different from what I was doing.