Choose a starting face and remember it. Walk each face of a cell containing that face touching each face once much like you would a 3-cube.

Pick any adjoining cell and move into one of its faces from there, walk each of its faces saving the one opposite the face you started on for last.

From there you're on a shared face with the cell opposite your starting cell. Traverse this one in a similar manner to the last, but this time also visit the adjoining faces of each cell adjacent to the second cell you filled, before once again ending opposite the face you started on for this cell.

Now you're on a shared face with the final cell, opposite the face you started on. Walk around the remaining four faces and you're done.