A lattice structure is defined by a network of interconnected structural members whose architecture exhibits some degree of regularity. Although the overall architecture of a lattice may contain many members, its generation can be a simple process in which a unit cell composed of a small amount of members, in comparison to the overall structure, is mapped throughout the Euclidean space. However, finding the right lattice architecture in a vast search space that customizes the behavior of a design for a given purpose, subject to mechanical and manufacturing constraints, is a challenging task. In response to this challenge, this work investigates a Voronoi diagram-based tessellation of a body-centered cubic cell for applications in structural synthesis and computational design of 3D lattice structures. This work contributes by exploring how the Voronoi tessellation can be utilized to parametrically represent the architecture of a lattice structure and what the implications of the parametrization are on the optimization, for which a global direct search method is used. The work considers two benchmark studies, a cubic and a cantilever lattice structure, as well as the effect of isotropic and anisotropic material property models, stemming from applications to additive manufacturing. The results show that the proposed parameterization generates complex search spaces using only four variables and includes four different lattice structure types, a Kelvin cell, a hexagonal lattice, a diamond-core lattice structure, and a box-boom type lattice structure. The global direct search method applied is shown to be effective considering two different material property models from an additive manufacturing (AM) process.