The analysis of wave propagation in solids with complex microstructures, and local heterogeneities finds extensive applications in areas such as material characterization, structural health monitoring (SHM), and metamaterial design. Within continuum mechanics, sources of heterogeneities are typically associated to localized defects in structural components, or to periodic microstructures in phononic crystals and acoustic metamaterials. Numerical analysis often requires computational meshes which are refined enough to resolve the wavelengths of deformation and to properly capture the fine geometrical features of the heterogeneities. It is common for the size of the microstructure to be small compared to the dimensions of the structural component under investigation, which suggests multiscale analysis as an effective approach to minimize computational costs while retaining predictive accuracy. This research proposes a multiscale framework for the efficient analysis of the dynamic behavior of heterogeneous solids. The developed methodology, called Geometric Multiscale Finite Element Method (GMsFEM), is based on the formulation of multi-node elements with numerically computed shape functions. Such shape functions are capable to explicitly model the geometry of heterogeneities at sub-elemental length scales, and are computed to automatically satisfy compatibility of the solution across the boundaries of adjacent elements. Numerical examples illustrate the approach and validate it through comparison with available analytical and numerical solutions. The developed methodology is then applied to the analysis of periodic media, structural lattices, and phononic crystal structures. Finally, GMsFEM is exploited to study the interaction of guided elastic waves and defects in plate structures.