Biology employs modular organization at every scale: molecular building blocks make up living cells, specialized cells organize within tissues, and collections of tissues constitute organs. 3D-printed networks of picoliter-sized aqueous compartments interconnected by lipid bilayers form a powerful platform for building precisely patterned synthetic tissues. However, this technology has been limited to millimeter-sized networks, with slow fabrication times and lacking flexible design. Here, the authors apply modular design to construct modular synthetic tissues by assembling a wide range of 3D-printed building blocks. They use dedicated modules for storing and releasing reagents, performing logic operations, responding to magnetic fields, and encapsulating living cells. They build centimeter-sized synthetic tissues able to transmit electrical signals through thousands of interconnected compartments. They assemble hybrid tissues composed of both synthetic modules and modules containing living cells. Lastly, by incorporating mutant protein nanopores within the building blocks, they assemble modular synthetic tissues with electrical outputs that are modulated by the integration of chemical inputs.