The Hamiltonian approach to the calculation of vibrational modes of elastic cubic objects is exploited, including the effect of higher-order elastic constants. A technique is presented for solving the inverse problem, i.e., the determination of second- and third-order elastic constants for face cubic centered crystals by inversion of natural frequencies data. An unconstrained minimization algorithm, using gradient methods, is used to numerically solve the inverse problem. The method is applied to the case of Al alloys, since they are particularly suitable for applications of the acoustoelastic effect, which requires a knowledge of the third-order elastic constants. The efficiency and the accuracy of the method and the influence of measurement errors are discussed.