We use the time-dependent Gross-Pitaevskii equation to study the motion of a vortex ring produced by phase imprinting on an elongated cloud of cold atoms. Our approach models the experiments of Yefsah on 6Li in the Bose-Einstein- condensate regime where the fermions are tightly bound into bosonic dimers. We find ring oscillation periods which are much larger than the period of the axial harmonic trap. Our results lend further strength to Bulgac 's arguments (arXiv:1306.4266) that the "heavy solitons" seen in those experiments are actually vortex rings. We numerically calculate the periods of oscillation for the vortex rings as a function of interaction strength, trap aspect ratio, and minimum vortex ring radius. In the presence of axial anisotropies the rings undergo complicated internal dynamics where they break into sets of vortex lines, then later combine into rings. These structures oscillate with a similar frequency to simple axially symmetric rings. © 2013 American Physical Society.