Discrete dipole-exchange spin wave spectra of tangentially magnetized non-ellipsoidal (circular cylindrical and rectangular) magnetic dots are calculated. The dots are assumed to be flat single domain particles in which the in-plane dot sizes are much larger than the dot height (film thickness). The uniform distribution of the variable magnetization along the dot height is assumed. The spectrum. of spin waves with in-plane wave vectors is found from the solution of the Landau-Lifshitz equation and the magnetostatic Maxwell equations. The approximate spin-wave dispersion equation is obtained using the diagonal approximation in a simple analytical form similar to that of an infinite film. The spectrum of discrete spin wave modes in permalloy (FeNi) cylindrical and rectangular dots is calculated and compared with experiment.