The rates of electron scattering by acoustic phonons in a rectangular quantum wire embedded into another material are calculated in the framework of the Fermi golden rule. Both intrasubband and intersubband electron scattering by acoustic phonons are considered. It has been shown that due to uncertainty of momentum conservation in quasi-one-dimensional systems the acoustic-phonon scattering becomes essentially inelastic in contrast to that in bulk materials. The acoustic-phonon scattering rate in quantum wires increases with the decrease of cross section of the wire and is much greater than that in bulk materials. It is shown that the correct treatment of inelasticity leads to a nonmonotonic dependence of the emission rate on electron energy and to the disappearance of the divergency of the acoustic-phonon scattering rate at the bottom of each subband. Therefore, the scattering time averaged over the distribution function exceeds considerably that calculated using elastic and quasielastic approaches where the scattering rate is divergent. We demonstrate that electron mobility at temperatures less than 100 K calculated within elastic or quasielastic approximations is greatly underestimated.