Radiation pressure and photon momentum in negative-index media are no different than their counterparts in ordinary (positive-index) materials. This is because the parameters responsible for these properties are the admittance √ε/μ and the group refractive index ng of the material (both positive entities), and not the phase refractive index n = √εμ, which is negative in negative-index media. One approach to investigating the exchange of momentum between electromagnetic waves and material media is via the Doppler shift phenomenon. In this paper we use the Doppler shift to arrive at an expression for the radiation pressure on a mirror submerged in a negative-index medium. In preparation for the analysis, we investigate the phenomenon of Doppler shift in various settings, and show the conditions under which a so-called "inverse" Doppler shift could occur. We also argue that a recent observation of the inverse Doppler shift upon reflection from a negative-index medium cannot be correct, because it violates the conservation laws.