In the context of two-dimensional discrete elasticity, we study the propagation of elastic waves in a solid plane made of a metamaterial with a microstructure engineered according to the Heuristic Molecules approach proposed by (Casolo, 2021). The RBSM-HM approach allows us to design topologies consisting of rigid elements (Heuristic Atoms) and elastic connections (to form Heuristic Molecules) in order to obtain isotropic and centrosymmetric assemblies at the macroscale in the static and linear-elastic fields, with micro-polar properties. The single periodic cell includes two types of internal first-order-neighbour elastic connections: blue (axial and non-central) springs and magenta (shear) springs. Different calibrations of elastic connections allow for assemblies that can be traced back to different materials at the macroscale, starting from a standard Cauchy continuum and arriving at the case of a micro-polar isotropic Cosserat continuum analysed in this study. Extending our study to the dynamic field reveals some characteristics of elastic wave propagation that are influenced by topology and micro-polarity (non-centrality of axial connections). These effects are studied in the presence of both affine and non-affine dynamic deformation fields. These phenomena, combined with particular characteristics of the wave incident on the discrete medium, can lead to dispersion, reflections, refractions, and anomalies in the mechanical response of the medium in the dynamic field. A rectangular plate is used as a case study for a parametric numerical analysis campaign using a specific code implemented in MatLAB. The objective is to understand the relationships between the spectral characteristics of the wave and those specific to the microstructure, highlighting the peculiarities in the elasto-dynamic behaviour of the lamina.