We present the Empirical Interscale Finite Element Method (EIFEM), a novel FE technology for modeling multiscale heterogeneous structures in both dynamic and static analyses. The method combines a Localized-Lagrange multiplier domain decomposition technique with dimensional hyperreduction at the subdomain level. However, in contrast to other multiscale techniques, such as the FE2, the proposed method is based on an 'interscale' variational framework which directly relates coarse-scale nodal internal forces with fine-scale stresses, thereby avoiding the typical nested sequence of local/global problems. This distinguishing feature, together with hyperreduction schemes for nodal internal and external body forces, greatly facilitates its implementation in existing FE codes for solid elements. For this, one only has to change the location and weights of the integration points, and replace polynomial-based finite element matrices with empirical operators, i.e., operators computed from the information obtained, in an offline stage, in appropriately designed computational experiments. We demonstrate that the elements resulting from this formulation do not suffer volumetric locking when dealing with nearly-incompressible materials, and that they can naturally deal with non-matching fine-scale grids as well as curved structures, for both linear and nonlinear regimes (in small strains). Lastly, we demonstrate that, for periodic structures, this method converges upon mesh refinement to the solution furnished by classical first-order computational homogenization. Thus, albeit the method does not presume scale separation, it can represent this limiting case by using sufficiently small coarse-scale elements.