Spin-wave modes in Ni80Fe20 thin-film antidot lattices are investigated using micromagnetic simulations and a semianalytical theoretical approach. The simulations reveal a rich eigenmode spectrum consisting of edge and center modes. We find both spatially localized and spin waves extending over many unit cells. To classify the different types of modes and to analyze the microscopic properties, we adapt a semianalytical approach. We show how to reduce the two-dimensional problem of the antidot lattice to a one-dimensional problem if certain high-symmetry axes are considered. For lattices of unit-cell lengths ranging from 200 to 1100 nm, we find that the characteristic mode eigenfrequencies can be correlated with both local inhomogeneities of the demagnetization field and specific wave vectors caused by geometry-imposed mode quantization conditions. We compare our results with recently published experimental data and discuss the crossover from dipolar to exchange-dominated spin waves. Moreover, we simulate propagation of spin waves and find a preferred axis of propagation perpendicular to the external magnetic field.