Foldable metamaterials are often used to create deployable structures with unique mechanical properties, such as programmable stiffness, reconfigurable topology, and large volumetric change. These metamaterials are constructed from individually transformable unit cells that are interconnected to form shape-changing macrostructures. In this work, we explore an origami-inspired stellated octahedron unit cell with three-dimensional flattenability and propose a kinematic characterization method based on the unique geometry of a unit sphere passing through a surface plane. Based on this unit cell, we propose a new class of reconfigurable 3D lattice metamaterials that can fold flat across multiple dimensions. We establish the energy profile of a single unit as a function of spring forces acting on the design, providing a framework for engineers to tailor the energetic behavior of the system on both an individual and macroscopic scale. From these base kinematics, we introduce a set of connective topologies to create metamaterials with varying shape-changing properties in a three-dimensional lattice. We also introduce a method for generating valid 3D tessellations and enforcing constraints of local and global connectivity across modules.