Interphase microtubule organization is critical for cell function and tissue architecture. In general, physical mechanisms are sufficient to drive microtubule organization in single cells, whereas cells within tissues are thought to utilize signalling mechanisms. By improving the imaging and quantitation of microtubule alignment within developing Drosophila embryos, here we demonstrate that microtubule alignment underneath the apical surface of epithelial cells follows cell shape. During development, epidermal cell elongation and microtubule alignment occur simultaneously, but by perturbing cell shape, we discover that microtubule organization responds to cell shape, rather than the converse. A simple set of microtubule behaviour rules is sufficient for a computer model to mimic the observed responses to changes in cell surface geometry. Moreover, we show that microtubules colliding with cell boundaries zip-up or depolymerize in an angle-dependent manner, as predicted by the model. Finally, we show microtubule alignment responds to cell shape in diverse epithelia.