Our Sun lies within 300 pc of the 2.7-kpc-long sinusoidal chain of dense gas clouds known as the Radcliffe Wave.1 The structure’s wave-like shape was discovered using 3D dust mapping, but initial kinematic searches for oscillatory motion were inconclusive.2–7 Here we present evidence that the Radcliffe Wave is oscillating through the Galactic plane while also drifting radially away from the Galactic Center. We use measurements of line-of-sight velocity8 for 12CO and 3D velocities of young stellar clusters to show that the most massive star-forming regions spatially associated with the Radcliffe Wave (including Orion, Cepheus, North America, and Cygnus X) move as if they are part of an oscillating wave driven by the gravitational acceleration of the Galactic potential. By treating the Radcliffe Wave as a coherently oscillating structure, we can derive its motion independently of the local Galactic mass distribution, and directly measure local properties of the Galactic potential as well as the Sun’s vertical oscillation period. In addition, the measured drift of the Radcliffe Wave radially outward from the Galactic Center suggests that the cluster whose supernovae ultimately created today’s expanding Local Bubble9 may have been born in the Radcliffe Wave.