Previous studies on elastic wave propagation in fractured media have demonstrated that a single planar fracture supports fracture interface waves and that two plane parallel fractures support fracture channel waves. Here, the results are presented for plane wave propagation through an infinite number of plane parallel fractures with equal fracture spacing and fracture stiffnesses. Analysis of the dispersion equations for this fractured system demonstrates that these waves exhibit symmetric and antisymmetric particle motions, degenerate to classical Rayleigh-Lamb plate waves when the fractures are completely open, and possess dispersive velocities that are functions of both the fracture stiffness and spacing. Time-frequency analysis performed on a series of laboratory ultrasonic transmission measurements on a fractured rock analog shows good agreement with the theoretical predictions.