Is a Radial Tube Safe in a Radial Tire ?

My MGB has wire wheels -- tubes are a necessity. All of the wire wheel classic cars run tubes, and if they have radial tires (all the non-trailer-queens), they run radial tubes. Nothing the matter with them, assuming they are the correct size. However, I have noticed tire shops generally don't want anything to do with either wire wheels or tubes. Poor training is my guess.