We are concerned with the efficient numerical approximation of solutions of the time-domain Maxwell system. Methods like the classical finite-difference time-domain method or finite integration techniques rely on the approximation of the electric and the magnetic field on two interlaced (Cartesian) grids respectively. Our method expands this idea to general triangular meshes by defining a dual grid using the barycentric subdivision of the primal mesh, resulting in a decomposition of each mesh into quadrilaterals. Contrary to previous approaches we use a Lagrangian polynomial basis with respect to tensor product integration points on the unit square which are subsequently re-mapped to the quadrilaterals of the reference element. This approach also provides a generalization of the method to three space dimensions and enables us to use lumped mass matrices which significantly increases the computational efficiency of our method. Numerical experiments underline the facts that the resulting algorithm provides converging, spurious-free solutions and is efficient, compared to competing methods.