An algorithm for computing surface skeletons on the face-centered cubic (fcc) grid and the body-centered cubic (bcc) grid is presented. The fcc grid and the bcc grid are three-dimensional grids where the voxels are rhombic do-decahedra and truncated octahedra, respectively. The DT is used to generate the set of centres of maximal balls (CMBs) which will be "anchor points" when constructing the skeleton. Simple points are used in order to make the skeleton topologically correct and CMBs to produce a fully reversible skeleton. Using only simple points and the CMBs generates a skeleton with a lot of branches. By using a set of additional conditions for removal and preservation of gridpoints, most of these branches are merged into surfaces. For comparison, the algorithm is also implemented for the cubic grid.