The shortest path passing on the surface of a polyhedron is called a geodesic path. A geodesic path of a polyhedron has a property that it becomes a single line segment on a development. A geodesic path is the shortest path and it mostly passes a small number of faces. We, however, consider a problem “is there a case that a geodesic path passes all faces of a polyhedron?” For this problem the answer is “yes”: we found that a regular tetrahedron has such a geodesic path. The next question is “what polyhedra have such geodesic paths?” We define a face-guard geodesic path (FGG path, for short) as a geodesic path connecting two points on a polyhedron and passing through all its faces, call a polyhedron that has an FGG path an FGG polyhedron, and try to characterize FGG polyhedra. For this new problem, we prove that there exists an FGG n-hedron for any integer \(n\ge 4\) , all tetrahedra and all triangular prisms with one exception are FGG polyhedra, and all cuboids and all regular polyhedra except regular tetrahedra are not FGG polyhedra.