The 3D organisation of cells determines tissue function and integrity, and changes dramatically in development and disease. Cell-based simulations have long been used to define the underlying mechanical principles. However, large computational costs have so far limited simulations to either simplified cell geometries or small tissue patches. Here, we present SimuCell3D, a highly efficient open-source program to simulate large tissues in 3D with subcellular resolution, growth, proliferation, extracellular matrix, fluid cavities, nuclei, and non-uniform mechanical properties, as found in polarised epithelia. Spheroids, vesicles, sheets, tubes, and other tissue geometries can readily be imported from microscopy images and simulated to infer biomechanical parameters. Doing so, we show that 3D cell shapes in layered and pseudostratified epithelia are largely governed by a competition between surface tension and intercellular adhesion. SimuCell3D enables the large-scale in silico study of 3D tissue organization in development and disease at an unprecedented level of detail.