Embodiments of modular building units (blocks) are disclosed that can be stacked to provide insulating concrete forms (ICFs). The blocks are formed of a foamed polymer material, such as foamed polyurethane. The blocks, when assembled together in an ICF, are configured to form one or more vertical columns of concrete and one or more horizontal beams of concrete intersecting the one or more columns, in a manner of post and beam construction. The blocks can be tightly interlocked with other blocks in a construction process. The disclosed blocks can include significant interior space for rebar and concrete to be inserted to construct the structure. Molds and methods for producing modular units are also disclosed. The molds and methods enable rapid production and removal of modular building units from the molds. Methods of constructing structures using the modular building units are also disclosed.