Elastic bulk materials support longitudinal and transverse waves such that the former travels faster in most cases. Anomalous polarization is the case when a transverse wave travels faster, allowing us to engineer the wave propagation via wave steering, scattering control, and mode conversion, which has critical applications in vibration mitigation and ultrasonic imaging. However, realizable materials that exhibit anomalous polarization are rarely found in nature, and architected materials that exhibit this property have only been demonstrated in 2D. In this article, we present 3D auxetic periodic lattice materials that support anomalous wave polarization. Through finite element analysis, we show that these lattices can switch between normal and anomalous behavior via simple geometry changes. We confirm the elasticity condition and qualitatively discuss the guidelines to design lattice materials that support anomalous polarization along a specific wave propagation direction. We show the ability to control the anomalous wave propagation direction by modifying the lattice geometry. Further, we numerically demonstrate mode conservation, deceleration, and acceleration of an incident wave using a material that exhibits anomalous wave polarization. These demonstrations show the potential application of such materials in nondestructive evaluation and medical imaging.