A solution method is given to the problem of plane wave propagation through one- and two-dimensional platonic arrays. The problem is formulated in terms of boundary integral equations and a solution is constructed using boundary element methods. Previous work has been restricted to simple geometries such as circles, pins and squares, and here the framework is extended to consider scatterers of arbitrary shape subject to clamped-edge boundary conditions at the edge. This is done by constructing scattering matrices for a single grating and using Bloch's theorem to form an eigenvalue problem which connects the grating problem to the array problem. The associated eigenvalues then permit the construction of band surfaces which reveal the flexural wave filtering capabilities of different geometries, as well as the behaviour of Bloch waves within the array. Multiple geometries are investigated and the first band surfaces are computed for these specific cases.