Negative Poisson's ratio metamaterials have significant application potential in aerospace, biomedical engineering, and flexible electronics due to their anomalous mechanical behavior of expanding laterally under tension and contracting laterally under compression. However, existing research predominantly focuses on 2D or 3D isotropic designs, which struggle to meet practical engineering demands for direction-dependent material properties. This paper proposed a density-based topology optimization method for designing 3D orthotropic negative Poisson's ratio metamaterials. By constructing a novel multi-objective optimization function combined with homogenization theory, negative Poisson's ratio characteristics in three orthogonal directions for unit cell structures was achieved. First, based on the SIMP (solid isotropic material with penalization) material interpolation model, geometric constraints were introduced to ensure unit cell symmetry while effectively reducing computational scale. Second, a new objective function and optimization model were established through penalty functions. Finally, the equivalent mechanical properties were calculated using finite element homogenization under periodic boundary conditions, with design variables updated through sensitivity analysis. Numerical examples demonstrate that the optimized unit cells exhibit negative Poisson's ratio behavior in all three principal directions. This study provides a theoretical foundation for the controllable fabrication of 3D orthotropic auxetic metamaterials and expands the design and application scope of mechanical metamaterials.