We use the theory of periodic structures, full-wave electromagnetic, and microwave circuit simulations to explain the resonant modes that propagate in metallic grids having rectangular unit cells constructed over a ground plane. We show that these metallic grids can support two types of resonant modes that have rectangular and hyperbolic isofrequency dispersion contours. By exploiting the spatial dispersion properties of these modes, a microwave 3GHz/6GHz harmonic splitter and a highly selective 5.8GHz/6.2GHz diplexer are designed and simulated. Furthermore, we provide experimental results for the diplexer and for the harmonic splitter, synthesized in microstrip technology. The proposed metallic grids utilize continuous unloaded transmission-line segments thus leading to spatial-filtering devices that are easy to fabricate and are scalable to terahertz frequencies and beyond.