We present a spectral scheme specially developed to simulate the delamination of thin films subjected to dynamic anti-plane shear loading conditions. The numerical scheme is based on an exact spectral representation of the elastodynamic relations between the interface traction stress, the interface displacement and the transient traction applied along the surface of the thin film. The formulation incorporates the contribution from all the wave reflections taking place in the film and the effect of the material mismatch between the film and the substrate, both of which are assumed to be linearly elastic. Its implementation involves an explicit time stepping scheme with, for each time step, the use of Fast Fourier Transform (FFT) to link the spatial and spectral domains, and the computation of a convolution over the past displacement and stresses history. A low-pass filter is also used in order to improve the stability of the method without affecting the precision of the results. We apply the developed scheme to thin film delamination problems involving non-propagating and propagating interface cracks. In the non-propagating case, special focus is placed on extracting the time-dependent stress intensity factor and on relating its evolution to the complex wave reflection events taking place in the thin film. In the propagating crack problem, we investigate the effect of the wave reflections off the film surface on the subsonic and intersonic crack motion.