A continuum model for elastic periodic composites with fibers of finite dimensions is formulated. The proposed theory provides the dispersion relations for harmonic waves in the three-dimensional composite. Dispersion relations of propagating waves normal to the fibers in long-fiber composites, and normal to the layering in periodically bilaminated composites are obtained as special cases. The formulation is given for composites with elastic anisotropic constituents. The predicted acoustical dispersion curves are checked with the exact elasticity solution for laminated composite, and with solutions based on variational principles for fibrous composites. The effect of the strong anisotropy of the fibers in a graphite/epoxy composite is illustrated.