Particle-in-cell (PIC) simulations of relativistic flowing plasmas are of key interest to several fields of physics (including, e.g., laser-wakefield acceleration, when viewed in a Lorentz-boosted frame) but remain sometimes infeasible due to the well-known numerical Cherenkov instability (NCI). In this article, we show that, for a plasma drifting at a uniform relativistic velocity, the NCI can be eliminated by simply integrating the PIC equations in Galilean coordinates that follow the plasma (also sometimes known as comoving coordinates) within a spectral analytical framework. The elimination of the NCI is verified empirically and confirmed by a theoretical analysis of the instability. Moreover, it is shown that this method is applicable both to Cartesian geometry and to cylindrical geometry with azimuthal Fourier decomposition.