The morphogenetic patterning that generates three-dimensional (3D) tissues requires dynamic concerted rearrangements of individual cells with respect to each other. We have developed lithography-based 3D culture models that recapitulate the architecture of epithelial ductal trees, enable micrometer-resolution control of tissue geometry and microenvironment, and provide quantitative 4D data in a physiologically relevant context. This approach has revealed that patterns within developing tissues can emerge from at least three orthogonal mechanisms: biochemical gradients, tissue mechanics, and differential cell motility. I will discuss how we combine engineered tissues and computational models to uncover and dissect the relative roles of each of these mechanisms.