The study of intercellular signalling networks requires organotypic microscale systems that facilitate the culture, conditioning and manipulation of cells. Here, we describe a reconfigurable microfluidic cell-culture system that facilitates the assembly of three-dimensional tissue models by stacking layers that contain preconditioned microenvironments. By using principles of open and suspended microfluidics, the Stacks system is easily assembled or disassembled to provide spatial and temporal manoeuvrability in two-dimensional and three-dimensional assays of multiple cell types, enabling the modelling of sequential paracrine-signalling events, such as tumour-cell-mediated differentiation of macrophages and macrophage-facilitated angiogenesis. We used Stacks to recapitulate the in vivo observation that different prostate cancer tissues polarize macrophages with distinct gene-expression profiles of pro-inflammatory and anti-inflammatory cytokines. Stacks also enabled us to show that these two types of macrophages signal distinctly to endothelial cells, leading to blood vessels with different morphologies. Our proof-of-concept experiments exemplify how Stacks can efficiently model multicellular interactions and highlight the importance of spatiotemporal specificity in intercellular signalling.