A new method for deriving anisotropic atomic dipole polarizability tensors for predicting molecular polarizabilities is presented. The procedure uses the relationships between atomic and molecular multipoles to rigorously derive the atomic polarizability tensors from quantum mechanically computed molecular dipole-quadrupole polarizabilities and their derivatives with respect to molecular deformations. The resulting atomic polarizabilities have been tested for the water molecule at several ab initio levels of approximation by comparing the summed atomic polarizabilities with the conventional ab initio molecular results and with experiment. Good agreement was found between molecular polarizabilities found using the two methods. The model is also shown to give interaction energies between a water molecule and a point charge that accurately reproduce ab initio results. A similar comparison of the atomic polarizability tensors is shown for the N-methylacetamide molecule. For most atom types, the atomic polarizability tensors were found to be significantly anisotropic, indicating that this method can significantly improve on the isotropic approximation for predicting molecular properties that depend strongly on polarization effects.