  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 et. al. [Nature \textbf{499},
426] on $^6$Li in the BEC 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
et. al.'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.
