In ferromagnetic thin films, broken inversion symmetry and spin-orbit coupling give rise to interfacial Dzyaloshinskii-Moriya interactions. Analytic expressions for spin-wave properties show that the interfacial Dzyaloshinskii-Moriya interaction leads to nonreciprocal spin-wave propagation, i.e., different properties for spin waves propagating in opposite directions. In favorable situations, it can increase the spin-wave attenuation length. Comparing measured spin-wave properties in ferromagnet/normal metal bilayers and other artificial layered structures with these calculations could provide a useful characterization of the interfacial Dzyaloshinskii-Moriya interactions.