We have developed a theory that describes the spin-wave spectra of ferromagnetic films with Dzyaloshinskii-Moriya interactions. In agreement with recent experiments (Zakeri et al 2010 Phys. Rev. Lett. 104 137203), we demonstrate that the spin-wave dispersion relation is asymmetric with respect to wave vector inversion for a variety of ferromagnetic films with Dzyaloshinskii-Moriya interactions and different crystallographic classes. It is also predicted that, for non-zero wave vectors, the resonance frequency and resonance field can increase or decrease depending on the spin-wave vector orientation. We provide explicit formulas for the spin-wave dispersion relation and its asymmetry, as well as for the dynamic susceptibility for a film under microwave excitation, that can be used to understand ferromagnetic resonance as well as Brillouin light scattering experiments in these classes of magnetic thin films.