We study the nature of rapidly star-forming galaxies at z= 2 in cosmological hydrodynamic simulations, and compare their properties to observations of submillimetre galaxies (SMGs). We identify simulated SMGs as the most rapidly star-forming systems that match the observed number density of SMGs. In our models, SMGs are massive galaxies sitting at the centres of large potential wells, being fed by smooth infall and gas-rich satellites at rates comparable to their star formation rates (SFRs). They are not typically undergoing major mergers that significantly boost their quiescent SFR, but they still often show complex gas morphologies and kinematics. Our simulated SMGs have stellar masses of M*∼ 1011−11.7 M⊙, SFRs of ∼180–500 M⊙ yr−1, a clustering length of ∼10 h−1 Mpc and solar metallicities. The SFRs are lower than those inferred from far-infrared (far-IR) data by ∼×3, which we suggest may owe to one or more systematic effects in the SFR calibrations. SMGs at z= 2 live in ∼1013 M⊙ haloes, and by z= 0 they mostly end up as brightest group galaxies in ∼1014 M⊙ haloes. We predict that higher M* SMGs should have on average lower specific SFRs, less disturbed morphologies and higher clustering. We also predict that deeper far-IR surveys will smoothly join SMGs on to the massive end of the SFR–M* relationship defined by lower mass z∼ 2 galaxies. Overall, our simulated rapid star-formers provide as good a match to available SMG data as merger-based scenarios, offering an alternative scenario that emerges naturally from cosmological simulations.