Devices based on piezoelectric semiconductors (PSCs) have recently received particular attention due to their wide bandgap where strain energy band engineering under both static and time-harmonic deformations is the key. In this paper, we investigate and characterize the elastic waves propagating in an anisotropic n-type PSC plate. To achieve our goals, we first introduce the new notations for the extended displacements, stresses, strains, and modulus to arrive at a mathematically elegant extended Stroh formalism. Then, the elastic wave problem is converted into a linear eigenvalue system from which the extended displacements and stresses are expressed in terms of the eigenvalues and eigenvectors. Finally, making use of the boundary conditions on the top and bottom surfaces of the plate, wave dispersion and attenuation are derived analytically. Numerical examples are presented to systematically study the effect of the surface boundary condition, steady-state carrier density, plate thickness, and biasing electric field on the wave speed and attenuation of both shear horizontal and Lamb waves in the transversely isotropic ZnO PSC plate. Some interesting characteristics of the elastic waves observed in this paper could be helpful as theoretical guidance when designing PSC-based devices.