We are concerned with the efficient numerical approximation of solutions of the time-domain linear 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 tetrahedral meshes by defining a dual grid using the barycentric subdivision of the primal mesh, resulting in a decomposition into hexahedra. Contrary to previous approaches we use Lagrangian polynomial basis functions with respect to tensor product integration points on the unit cube that are subsequently re-mapped to the hexahedra which form the physical elements of the primal and dual grids respectively. This approach provides generically computable, stable high-order bases. We use leap frog time-stepping (or high-order variants thereof) for the time discretization which only requires the application of the discrete differential operators and the inverse mass matrices in each time step. Using lumped mass matrices with respect to the defining points of the basis functions leads to block diagonal mass matrices with block-size independent of the polynomial degree. Numerical experiments underline the facts that the resulting algorithm provides converging, spurious-free solutions and is efficient, compared to competing methods.