The interaction of anti-plane elastic SH waves with a periodic array of interface cracks in a one-dimensional phononic crystal is analyzed in this paper.A perfect periodic structure without interface cracks, a single and periodic distribution of cracks at the center interface are studied and the transmission displacement coefficients are obtained based on the transfer matrix method, Fourier integral transform and singular integral equation techniques.Compared with the corresponding perfect periodic structure without interface cracks, new phenomena are found in the phononic crystals with a single and periodic array of interface cracks.