Binary black holes radiate linear momentum in gravitational waves as they merge. Recoils imparted to the black-hole remnant can reach thousands of km/s, thus ejecting black holes from their host galaxies. We exploit recent advances in gravitational waveform modeling to quickly and reliably extract recoils imparted to generic, precessing, black hole binaries. Our procedure uses a numerical-relativity surrogate model to obtain the gravitational waveform given a set of binary parameters, then from this waveform we directly integrate the gravitational-wave linear momentum flux. This entirely bypasses the need of fitting formulae which are typically used to model black-hole recoils in astrophysical contexts. We provide a thorough exploration of the black-hole kick phenomenology in the parameter space, summarizing and extending previous numerical results on the topic. Our extraction procedure is made publicly available as a module for the Python programming language named surrkick. Kick evaluations take ∼0.1s on a standard off-the-shelf machine, thus making our code ideal to be ported to large-scale astrophysical studies.