Numerical calculations were performed for the melting process in porous media around a hot surface with conducting fins (fins with a large thermal-conductivity). The effect of the orientation of the hot surface on the effectiveness of the attached fins was examined. Three orientations, that is, the melting from below, from the side and from above, were selected as typical ones. Timewise variations of a temperature distribution, a velocity distribution, an interface shape, a mean Nusselt number and a melting fraction were compared. As a result, the temperature and flow fields became very complicated due to the attached conducting fins, especially in the cases of melting from below and from the side. The mean Nusselt number, (Nu)wBAR, over the hot surface for the melting from below was kept large for a long time, and varied irregularly with time. The value of (Nu)wBAR for the melting from above decreased monotonously with time. The melting rate was the largest for the melting from below, and the smallest for the melting from above, while the effect of the fins appeared relatively larger for the melting from above.