Architected materials are designed with specific configurations that offer enhanced properties, making them ideal for addressing various challenges in materials science, architecture, aerodynamics, and mechanical engineering. Their unique quality, coupled with the ability to tailor mechanical properties in every direction, renders them highly suitable for industries like aerospace, automotive, marine, and construction. However, the application of architected materials depends on the development of accurate models to understand the complex relationship between microstructure characteristics and macroscopic behavior. Despite the proposal and discussion of numerous analytical and numerical methods in recent years, very few studies have derived explicit formulas for effective mechanical properties. This paper contributes to this underexplored area by presenting a mathematical formulation and modeling technique for the effective elastic moduli of a three-dimensional orthotropic hexatruss lattice. The analytical relations we have established, validated through comprehensive experimental tests on a 3D-printed lattice, demonstrate the significant impact of lattice parameters on macroscopic properties. Practically, our results could simplify the process of parametric optimization for architectured materials, offering a less resource-intensive approach to optimization since parameter changes do not necessitate lattice regeneration.