In the context of acoustics, single polarization electromagnetism and elastic plates we consider microstructured media that have an underlying periodic structure and we develop an asymptotic continuum model that captures the essential microstructural behaviour entirely in a macroscale setting. The asymptotics are based upon a two-scale approach and are valid even at high frequencies when the wavelength and microscale length are of the same order. The general theory is illustrated via one- and two-dimensional model problems that can have zero-frequency stop bands that preclude conventional averaging and homogenization theories. Localized defect modes created by material or shape variations are also modelled using the theory and compared to numerical simulations.