  Using the variational approximation(VA) and direct simulations, we find
stable 2D and 3D solitons in the self-attractive Gross-Pitaevskii equation
(GPE) with a potential which is uniform in one direction ($z$) and periodic in
the others (but the quasi-1D potentials cannot stabilize 3D solitons). The
family of solitons includes single- and multi-peaked ones. The results apply to
Bose-Einstein condensates (BECs) in optical lattices (OLs), and to spatial or
spatiotemporal solitons in layered optical media. This is the first prediction
of {\em mobile} 2D and 3D solitons in BECs, as they keep mobility along $z$.
Head-on collisions of in-phase solitons lead to their fusion into a collapsing
pulse. Solitons colliding in adjacent OL-induced channels may form a bound
state (BS), which then relaxes to a stable asymmetric form. An initially
unstable soliton splits into a three-soliton BS. Localized states in the
self-repulsive GPE with the low-dimensional OL are found too.
