In the recent experiment, up to 1014g (g: the Earths gravitational acceleration) of acceleration for He atom is obtained by using the linearly polarized lasers of peak intensities around 1016 W/cm2.In the present study, we perform the classical trajectory Monte Carlo simulation of the complex dynamics of accelerating neutral atoms with linearly or circularly polarized intense laser pulses.Our simulations involve the ion motion as well as the tunneling ionization and the scattering dynamics of valence electron in the combined Coulomb and electromagnetic fields, for both helium (He) and magnesium (Mg).We show that for He atoms,only linearly polarized lasers can effectively accelerate the atoms, while for Mg atoms, we find that both linearly and circularly polarized lasers can successively accelerate the atoms.The underlying mechanism is discussed and the sub-cycle dynamics of accelerating trajectories is investigated We have compared our theoretical results with a recent experiment.