  We develop the scheme of dispersion management (DM) for three-dimensional
(3D) solitons in a multimode optical fiber. It is modeled by the parabolic
confining potential acting in the transverse plane in combination with the
cubic self-focusing. The DM map is adopted in the form of alternating segments
with anomalous and normal group-velocity dispersion. Previously, temporal DM
solitons were studied in detail in single-mode fibers, and some solutions for
2D spatiotemporal "light bullets", stabilized by DM, were found in the model of
a planar waveguide. By means of numerical methods, we demonstrate that
stability of the 3D spatiotemporal solitons is determined by the usual
DM-strength parameter, $S$: they are quasi-stable at $ S<S_{0}\approx 0.93$,
and completely stable at $S>S_{0}$. Stable vortex solitons are constructed too.
We also consider collisions between the 3D solitons, in both axial and
transverse directions. The interactions are quasi-elastic, including periodic
collisions between solitons which perform shuttle motion in the transverse
plane.
