We introduce a framework for construction of non-separable multivariate splines that are geometrically tailored for general sampling lattices. Voronoi splines are B-spline-like elements that inherit the geometry of a sampling lattice from its Voronoi cell and generate a lattice-shift-invariant spline space for approximation in R^d. The spline spaces associated with Voronoi splines have guaranteed approximation order and degree of continuity. By exploiting the geometric properties of Voronoi polytopes and zonotopes, we establish the relationship between Voronoi splines and box splines which are used for a closed-form characterization of the former. For Cartesian lattices, Voronoi splines coincide with tensor-product B-splines and for the 2-D hexagonal lattice, the proposed approach offers a reformulation of hex-splines in terms of multi-box splines. While the construction is for general multidimensional lattices, we particularly characterize bivariate and trivariate Voronoi splines for all 2-D and 3-D lattices and specifically study them for body centered cubic and face centered cubic lattices.