The reflection and transmission spectrum of an individual grating, and repeated stacks of gratings, is computed using boundary integral methods for arrays of arbitrarily shaped cavities, with free-edge boundary conditions, that are periodically repeated in a thin elastic plate. The solution is found using a specially developed boundary element method coupled with an array Green's function. The computational code is tested against the solution obtained using multipoles which applies only to circular geometries but is shown to give very good agreement. The code is also validated using energy balance equations which includes the evanescent modes. A number of cavity shapes are investigated and results are compared against circular cavities with equivalent volumes.