We present a new computational method to accurately identify the elastic wave propagation modes and polarizations in periodic solid structures and metamaterials. The method uses the eigenvectors associated with each propagating wave solution to calculate the contribution of each translational and rotational component of the total mass-in-motion. We use this information to identify the dominant wave propagation mode by defining a relative effective modal mass vector. Then, we associate each wave solution with its correct polarization by defining a polarization factor that quantifies the relative orientation between the wave propagation and lattice motion directions and provides a positive numerical value between 0 (pure S-wave) and 1 (pure P-wave). Further, we suggest a graphical representational scheme for easier visualization of the wave polarization within traditional dispersion plots. To validate the method, we compare our predictions against previously published results for various elastodynamic problems. Finally, we use the proposed method to analyze the effect of various lattice and structural parameter perturbations on the elastic wave propagation and polarized bandgap behavior of a square planar beam lattice. Our analysis reveals the emergence of previously unobserved dynamic characteristics, including various polarized bandgaps, fluid-like behavior, and ultralow-frequency SH- and SV-bandgaps that extend to 0 Hz. Our proposed method provides an alternative computational approach to the typically employed visual mode inspection technique and provides a robust method for analyzing the elastic wave response of periodic solid media.