Two-dimensional materials are emerging as a promising platform for ultrathin channels in field-effect transistors. To this aim, novel high-mobility semiconductors need to be found or engineered. Although extrinsic mechanisms can in general be minimized by improving fabrication processes, the suppression of intrinsic scattering (driven, for example, by electron-phonon interactions) requires modification of the electronic or vibrational properties of the material. Because intervalley scattering critically affects mobilities, a powerful approach to enhance transport performance relies on engineering the valley structure. We show here the power of this strategy using uniaxial strain to lift degeneracies and suppress scattering into entire valleys, dramatically improving performance. This is shown in detail for arsenene, where a 2% strain stops scattering into four of the six valleys and leads to a 600% increase in mobility. The mechanism is general and can be applied to many other materials, including in particular the isostructural antimonene and blue phosphorene.