  We study ultracold bosonic atoms with the synthetic three-dimensional
spin-orbit (SO) coupling in a cubic optical lattice. In the superfluidity
phase, the lowest energy band exhibits one, two or four pairs of degenerate
single-particle ground states depending on the SO-coupling strengths, which can
give rise to the condensate states with spin-stripes for the weak atomic
interactions. In the deep Mott-insulator regime, the effective spin Hamiltonian
of the system combines three-dimensional Heisenberg exchange interactions,
anisotropy interactions and Dzyaloshinskii-Moriya interactions. Based on Monte
Carlo simulations, we numerically demonstrate that the resulting Hamiltonian
with an additional Zeeman field has a rich phase diagram with spiral, stripe,
vortex crystal, and especially Skyrmion crystal spin-textures in each xy-plane
layer. The obtained Skyrmion crystals can be tunable with square and hexagonal
symmetries in a columnar manner along the z axis, and moreover are stable
against the inter-layer spin-spin interactions in a large parameter region.
