Periodic photonic structures enable precise control over the light-matter interaction through band-structure engineering. Certain lattice geometries support dispersionless flat bands, characterized by vanishing group velocity and diverging density of states, creating opportunities for applications such as slow light, nonlinear optical processes and control of photoluminescence. However, flat bands have not yet been reported in systems with long-range radiative coupling between lattice sites. Here, we show that lattices consisting of superposed equispaced one-dimensional chains exhibit flat bands with a purely diffractive origin, with the energies and angles of the flat bands determined by the geometry of the lattice and its unit cell. The flat bands extend over all angles, can have linewidths on the order of a few nanometers, and are linearly polarized. We experimentally observe flat bands at the predicted energies in lattices of gold nanoparticles at near-infrared frequencies using Fourier spectroscopy. Our results provide a general and efficient design strategy for lattices with flat, polarized bands with potential applications in flat-band lasing, enhanced light-matter interactions, and controlling the emission or absorption of electromagnetic radiation across a broad spectral range.