It is now routine to make Bose-Einstein condensates of polaritons with long enough lifetime and low enough disorder to travel ballistically for hundreds of microns in quasi-one-dimensional (1D) wires. We present observations of a nonequilibrium polariton condensate injected at one point in a quasi-1D ring, with a well-defined initial velocity and direction. A clear precession of the circular polarization is seen, which arises from an effective spin-orbit-coupling term in the Hamiltonian. Our theoretical model accurately predicts the experimentally observed behavior and shows that “zitterbewegung” behavior plays a role in the motion of the polaritons.