  We predict the existence of spatially localized nontrivial vortex states of a
Bose-Einstein condensate with repulsive atomic interaction confined by a
three-dimensional optical lattice. Such vortex-like structures include planar
vortices, their co- and counter-rotating bound states, and distinctly
three-dimensional non-planar vortex states. We demonstrate numerically that
many of these vortex structures are remarkably robust, and therefore can be
generated and observed in experiment.
