In this paper we prove the convergence of the law of grid-valued random walks, which can be seen as time-space Markov chains, to the law of a general diffusion process. This includes processes with sticky features, reflecting or absorbing boundaries and skew behavior. We show that for an arbitrary grid, the convergence occurs at a rate of order 1/4 in terms of the maximum cell size of the grid for any p-Wasserstein distance. We also show that it is possible to achieve convergence rates of order 1/2 if the grid is adapted to the speed measure of the diffusion, which is optimal. This result allows us to set up asymptotically optimal convergence schemes for general diffusion processes. Finally, we give several examples where the quantities that determine the law of the random walk are non-tractable or semi-tractable and where the diffusion it approximates exhibits various singular features.