The calculation of phonon dispersion for crystalline solids with r atoms in a unit cell requires solving a 3r-dimensional eigenvalue problem. In this paper we propose a simplified approach to lattice dynamics which yields approximate analytical expressions and accurate numerical solutions to phonon dispersion without solving the eigenvalue problem. This is accomplished by making coordinate transformations to the normal modes of the isolated unit cell, which are extended over the entire crystal by Fourier transformations, so each phonon branch is labelled by the irreducible representations of the symmetry group of the unit cell from which the atomic displacements can be readily identified. The resulting dynamical matrix can be analyzed perturbatively, with the diagonal elements as the zeroth order matrix and the off-diagonal elements as the perturbation. The zeroth-order matrix provides approximate analytical expressions for the phonon dispersions, the first-order terms vanish, and the higher-order terms converge to the exact solutions. We describe the application of this method to a one-dimensional diatomic chain, graphene, and hexagonal closepacked zirconium. In all cases, the zeroth-order solution provides reasonable approximations, while the second-order solutions are close to the exact dispersion curves. This approach provides insight into the lattice dynamics of crystals, molecular solids and Jahn-Teller systems, while significantly reducing the computational cost.