Grand canonical Monte Carlo simulations were carried out to examine phase transitions of rigid lattice objects. The objects studied are coarse representations of planar, spherical, cubic, and cylindrical shapes on a simple cubic lattice. Multihistogram reweighting methods were used to generate the phase diagrams; critical points were determined using finite-size scaling methods. Spheres and capped cylinders exhibit a first-order transition from fluid to crystalline phases in the athermal limit. Small cubes and planar structures show continuous order-disorder transitions. For a six-armed cross object, there is evidence of a metastable vapor-liquid-phase transition lying within the fluid-solid-phase envelope.