Cassini magnetic field observations show that few-nT oscillations near the planetary rotation period, first observed in Pioneer-11 and Voyager-1 and -2 fly-by data, are essentially ubiquitous in Saturn's magnetosphere, though their character differs between the quasi-dipolar ring current region and the dawn tail. Examination of data from the ring-current region shows, however, that the observed oscillation period is not fixed at the planetary period, but has smaller values on the inbound pass of the spacecraft, increasing to larger values at and beyond periapsis. These variations are shown to be consistent with the Doppler shifts due to spacecraft motion expected in a model in which the wave phase fronts rotate with the planet, and radiate outward at a speed comparable with the equatorial Alfv¿n speed.