Dynamic Mode III interfacial fracture in a dissimilar square-cell lattice, composed of two contrasting mass-spring lattice half-planes joined at an interface, is considered. The fracture, driven by a remotely applied load, is assumed to propagate at a constant speed along the interface. The choice of the load allows the solution of the problem to be matched with the crack tip field for a Mode III interfacial crack propagating between two dissimilar continuous elastic materials. The lattice problem is reduced to a system of functional equations of the Wiener–Hopf type through the Fourier transform. The derived solution of the system fully characterises the process. We demonstrate the existence of trapped vibration modes that propagate ahead of the crack along the interface during the failure process. In addition, we show as the crack propagates several preferential directions for wave radiation can emerge in the structured medium that are determined by the lattice dissimilarity. The energy attributed to the wave radiation as a result of the fracture process is studied and admissible fracture regimes supported by the structure are identified. The results are illustrated by numerical examples that demonstrate the influence of the dissimilarity of the lattice on the existence of the steady failure modes and the lattice dynamics.