We obtain the phase diagram of fully packed hard plates on the cubic lattice. Each plate covers an elementary plaquette of the cubic lattice and occupies its four vertices, with each vertex of the cubic lattice occupied by exactly one such plate. We consider the general case with fugacities s_{μ} for "μ plates," whose normal is the μ direction (μ=x,y,z). At and close to the isotropic point, we find, consistent with previous work, a phase with long-range sublattice order. When two of the fugacities s_{μ_{1}} and s_{μ_{2}} are comparable, and the third fugacity s_{μ_{3}} is much smaller, we find a spontaneously layered phase. In this phase, the system breaks up into disjoint slabs of width two stacked along the μ_{3} axis. μ_{1} and μ_{2} plates are preferentially contained entirely within these slabs, while plates straddling two successive slabs have a lower density. This corresponds to a twofold breaking of translation symmetry along the μ_{3} axis. In the opposite limit, with μ_{3}≫μ_{1}∼μ_{2}, we find a phase with long-range columnar order, corresponding to simultaneous twofold symmetry breaking of lattice translation symmetry in directions μ_{1} and μ_{2}. The spontaneously layered phases display critical behavior, with power-law decay of correlations in the μ_{1} and μ_{2} directions when the slabs are stacked in the μ_{3} direction, and represent examples of "floating phases" discussed earlier in the context of coupled Luttinger liquids and quasi-two-dimensional classical systems. We ascribe this remarkable behavior to the constrained motion of defects in this phase, and we sketch a coarse-grained effective field theoretical understanding of the stability of power-law order in this unusual three-dimensional floating phase.