Diffusive transport in media with discontinuous properties is a challenging problem that arises in many applications. This paper presents a novel analytical expression from the method of images to simulate diffusive processes, such as mass or thermal transport, in discontinuous media with generalized boundary conditions at the discontinuity interface relying upon the random walk method. The analytical expressions are used to formulate a generalization of the existing Skew Brownian Motion, HYMLA and Uffink's method, here named as GSBM, GHYMLA and GUM respectively, to handle generic interface conditions. The algorithms are tested by simulating transport in a bimaterial medium with piece-wise constant properties. The results indicate that the GUM algorithm provides the best performance in terms of accuracy and computational cost. The methods proposed can be applied for simulation of a wide range of differential problems.