An edge-unfolding of a polyhedron is a cutting of the polyhedron's surface along its edges so that its surface can be flattened into a single connected flat patch on the plane without any self-overlapping. A one-layer lattice polyhedron is a polyhedron of height one, whose surface faces are grid squares. We consider the edge-unfolding problem on several classes of one-layer lattice polyhedra with cubic holes. We propose linear-time algorithms for one-layer lattice polyhedra with rectangular external boundary and cubic holes, one-layer lattice polyhedra with cubic holes strictly enclosed by an orthogonally convex polygon, and one-layer lattice polyhedra with sparse cubic holes, respectively. The algorithms use two different novel techniques to cut the edges of cubic holes of the given polyhedron so that no self-overlapping can occur in the flattened patch. Our algorithms are the first algorithms especially designed to edge-unfold a polyhedron of genus greater than zero to a single connected flattened patch. We leave open the question whether any of these edge-cutting methods can be extended to edge-unfold general one-layer lattice polyhedra with cubic holes.