A variant of stepwise calculation of the elastic constants of a polymer composite, containing platelike multilayer clay particles as filler, has been suggested. First, the elastic characteristics of particles in the form of multilayer stacks with interlaminar galleries, enlarged by the intercalation of polymer, were determined according to the elasticity relations for layered media. Then, using the Mori-Tanaka theory of an equivalent medium, the elastic constants of the transversely isotropic representative structural element of a composite with a coplanar arrangement of platelike clay particles were calculated. Finally, the elastic constants of a composite with randomly oriented particles were found by using the method of orientational averaging of elastic properties of the structural element of the composite.The effect of the number of monolayers in clay particles on the elastic constants of a composite was analyzed.