In the study of periodic problems it is natural and commonplace to use Fourier transforms to obtain explicit lattice Green’s functions in the form of multidimensional integrals. Considerable physical information is encapsulated within the Green’s function and our aim is to extract the behaviour near critical frequencies by creating connections with multiple-scale homogenisation methods recently applied to partial differential equations. We show that the integrals naturally contain two-scales, a short-scale on the scale of the lattice and a long-scale envelope. For pedagogic purposes we first consider the well-known two dimensional square lattice, followed by the three dimensional cubic lattice. The features we uncover, and the asymptotics, are generic for many lattice structures. Finally we consider a topical three dimensional example from structural mechanics showing dynamic anisotropy, that is, at specific frequencies all the energy is directed along specific characteristic directions.