Architected metamaterials are engineered material systems that provide unique mechanical or functional responses that cannot be achieved with naturally existing materials. They offer desirable properties like high strength-to-weight ratios, controlled acoustic and vibration response, or tunable thermal response. Current advances in manufacturing enable fabrication of these complex structures, driving their research interest and appeal for commercial applications. Our focus is on understanding the dynamic behavior of architected metamaterials with a periodic lattice structure, by investigating the effect of various lattice parameters and structural symmetries on the propagation of elastic waves. To this end, a new method to identify the wave polarization in the dispersion curves is proposed. The method uses eigenvector information obtained using finite element simulations to calculate the relative effective translational and rotational masses in motion along the different coordinate directions at each eigenfrequency. This information is then used to identify the structural vibration mode associated with the wave propagation at that frequency. Further, the wave polarization is identified using an angle of propagation calculated with reference to the lattice vector path. The method is validated using previously published data, and then used to demonstrate its effectiveness by identifying wave polarizations and bandgaps in strut and curved-based lattices -derived from the minimal surfaces Schwarz Primitive, Schwarz Diamond, and Gyroid, with alterations to their material and structural symmetries. The study revealed that symmetry modifications result in generation of additional directional and polarized bandgap, as well as geometry dependent-anomalous polarization. This development will enable engineers to design such structures for application-specific dynamic response and can also be applied to fields of soft metamaterials, photonics, or non-destructive testing techniques for periodic structures.