The unbinding properties of an interface near structured wedges are investigated by discrete models with short range interactions. The calculations demonstrate that interface unbinding takes place in two stages: (i) a continuous filling-like transition in the pure wedge-like parts of the structure, and (ii) a conclusive discontinuous unbinding. In 2D an exact transfer matrix approach allows us to extract the whole interface phase diagram and the precise mechanism at the basis of the phenomenon. The Metropolis Monte Carlo simulations performed in 3D reveal an analogous behavior. The emerging scenario allows us to shed new light onto the problem of wetting of geometrically rough walls.