It's my understanding that there is no C on most wiring, because there
isn't a need for one. (Ungrounded light switches don't have a common
wire; they just connect the hot wire to the light.) All of the
appliances are grounded so you only need the power wire and a wire for
each device. Connecting R (24VAC) to a appliance's wire completes the
circuit and allows the appliance to run.

If it has worked for years then, yes, something has gone wrong. I
would verify the wiring and then call the furnace guy to fix the
problem.

Mike