Despite the recognized role of liquid crystal microfluidics in generating programmable, self-organized and guided flow properties, to date, the flow behavior of LCs within curved channels remains unexplored. Using experiments and numerical simulations, we demonstrate that the curvature of microscale conduits allow programming of liquid crystal (LC) flows. Focusing on a nematic LC flowing through U- and L-shaped channels - two simple yet fundamental curved flow paths - with rectangular cross-section, our results reveal that the curvature of flow path can trigger transverse flow-induced director gradients. The emergent director field feeds back into the flow field, ultimately leading to LC flows controlled by the channel curvature. This curvature-mediated flow control, identified by polarizing optical microscopy and supported by the nematofluidic solutions, offers concepts in LC microfluidic valves, wherein the throughput distribution is determined by the Ericksen number and variations in local curvature. Finally, this work leverages curvature to amplify (suppress) LC transport through flow-aligned (homeotropic) regions emerging within channels with bends, in a programmable manner. Our results demonstrating the dependence of the dynamic flow-director coupling on the local curvature will have far-reaching ramifications in advancing the understanding of LC-based passive and active biological systems under real life geometrical constraints.