The asymptotic homogenization method (AHM) yields a two-scale procedure to obtain the effective properties of a composite material containing a periodic distribution of unidirectional circular cylindrical holes in a linear transversely isotropic piezoelectric matrix. The matrix material belongs to the symmetry crystal class 622. The holes are centered in a periodic array of cells of square cross sections and the periodicity is the same in two perpendicular directions. The composite state is antiplane shear piezoelectric, that is, a coupled state of out-of-plane shear deformation and in-plane electric field. Local problems that arise from the two-scale analysis using the AHM are solved by means of a complex variable method. For this, the solutions are expanded in power series of Weierstrass elliptic functions, which contain coefficients that are determined from the solutions of infinite systems of linear algebraic equations. Truncating the infinite systems up to a finite, but otherwise arbitrary, order of approximation, we obtain analytical formulas for effective elastic, piezoelectric, and dielectric properties, which depend on both the volume fraction of the holes and an electromechanical coupling factor of the matrix. Numerical results obtained from these formulas are compared with results obtained by the Mori–Tanaka approach. The results could be useful in bone mechanics.