The propagation of waves through microstructured media with periodically arranged
inclusions has applications in many areas of physics and engineering, stretching from photonic crystals
through to seismic metamaterials. In the high-frequency regime, modeling such behavior is
complicated by multiple scattering of the resulting short waves between the inclusions. Our aim
is to develop an asymptotic theory for modeling systems with arbitrarily shaped inclusions located
on general Bravais lattices. We then consider the limit of pointlike inclusions, the advantage being
that exact solutions can be obtained using Fourier methods, and go on to derive effective medium
equations using asymptotic analysis. This approach allows us to explore the underlying reasons for
dynamic anisotropy, localization of waves, and other properties typical of such systems, and in particular
their dependence upon geometry. Solutions of the effective medium equations are compared
with the exact solutions, shedding further light on the underlying physics. We focus on examples
that exhibit dynamic anisotropy as these demonstrate the capability of the asymptotic theory to pick
up detailed qualitative and quantitative features.