Francfort and Murat derived an explicit formula for the effective elasticity tensor of a multiply layered composite made from two isotropic elastic materials in prescribed proportion. For multiply layered composites with crystallographic symmetry, it is shown that these formulae can be represented as a group average over the crystallographic group. The special case of cubically symmetric elastic composites made by multiple layering is considered. This article determines precisely the set of elasticity tensors that correspond to these composites. Extremal property of laminar composites is then used to obtain new optimal bounds on the effective shear moduli for elastic composites with cubic symmetry.