The electronic transport behaviour of materials determines their suitability for technological applications. We develop an efficient method for calculating carrier scattering rates of solid-state semiconductors and insulators from first principles inputs. The present method extends existing polar and non-polar electron-phonon coupling, ionized impurity, and piezoelectric scattering mechanisms formulated for isotropic band structures to support highly anisotropic materials. We test the formalism by calculating the electronic transport properties of 16 semiconductors and comparing the results against experimental measurements. The present work is amenable for use in high-throughput computational workflows and enables accurate screening of carrier mobilities, lifetimes, and thermoelectric power.