This was the era where physicians were not only going blind on paperwork related to insurance, but also dealing with the red tape from the H.M.O.'s. Some physicians were running a service of direct payment from their patients. They had way more time on their hands to just practice medicine and see patients, and had far fewer overhead costs (no need for a staff of 8 to run your business - just an administrator to send out bills and deposit checks).
Never really took off widespread, because people generally still had to carry insurance to cover significant possibilities (injury, major illness), so the cost difference to them ended up being cheaper to just go ahead and carry full coverage and do what they've always done with physicians' offices that accepted insurance. The people who liked the cash-only practices were mostly just those who really liked that specific doctor and happened to be able to afford such a "luxury."
I could see this being the sort of thing that makes a comeback depending on the type of healthcare system that this country eventually goes to. In Canada, private practices like that do a fair amount of business with people who can afford them for certain services (and/or who don't want to wait).