After a brief description of the layer multiple scattering method as applied to phononic crystals, we present some results obtained by this method, relating to: crystals of polystyrene spheres in water; crystals of silica spheres in air; and crystals of steel spheres in polyester. We relate the transmission characteristics of slabs of these ma terials to the complex band structure of the corresponding infinite crystals. We emphasize aspects of the underlying physics which have not been discussed previously.