For a periodically driven quantum system, an effective time-independent Hamiltonian is derived with an eigenenergy spectrum, which in the regime of large driving frequencies approximates the quasienergies of the corresponding Floquet Hamiltonian. The effective Hamiltonian is evaluated for the case of optical lattice models in the tight-binding regime subjected to strong periodic driving. Three scenarios are considered: a periodically shifted one-dimensional (1D) lattice, a two-dimensional (2D) square lattice with inversely phased temporal modulation of the well depths of adjacent lattice sites, and a 2D lattice subjected to an array of microscopic rotors commensurate with its plaquette structure. In the 1D scenario, the rescaling of the tunneling energy, previously considered by Eckardt [Phys. Rev. Lett.PRLTAO0031-900710.1103/ PhysRevLett.95.260404 95, 260404 (2005)] is reproduced. The 2D lattice with well-depth modulation turns out as a generalization of the 1D case. In the 2D case with staggered rotation, the expression previously found in the case of weak driving by Lim [Phys. Rev. Lett.PRLTAO0031-900710.1103/PhysRevLett.100. 130402 100, 130402 (2008)] is generalized, such that its interpretation in terms of an artificial staggered magnetic field can be extended into the regime of strong driving.