Leaky waves are complex eigenmodes compatible with radiation in terms of their momentum, which can be supported by open waveguiding structures and can be effective at modeling radiative phenomena from such systems. Here, a “flatland” analog of leaky-wave radiation is introduced to describe a novel mechanism for in-plane radiation leakage that can occur in artificial or natural low-dimensional materials. Possible platforms are illustrated for its physical realization, in the form of suitably designed planar junctions of isotropic impedance surfaces that implement surface-wave or line-wave guiding structures. A simple and insightful semi-analytical model for the radiation mechanism is developed and validated against full-wave numerical simulations. These results provide a new tool for the advanced manipulation of surface waves at the nanoscale, which relies entirely on interface-optics effects, and can find interesting applications in the emerging field of polaritonics.