Current and planned low frequency radio telescopes (GMRT, LOFAR, MWA, LWA, eventually SKA) can probe non-thermal properties of the cosmic webs. The peaks in the diffuse, polarized emission from giant radio relics form a "gischt" upon structure formation shock waves which "illuminates" the large-scale magnetic field. We model the population of shock-accelerated relativistic electrons in high-resolution gas-dynamical simulations of galaxy clusters and the cluster magnetic field to show many more low-luminosity relics should be found than those seen so far, and that the relic luminosities and number counts strongly depend on the magnetic field, the cluster mass and the dynamical state, with multiple relics expected for each violently merging cluster. In the best cases, the radio observables enable us to extract unique physical properties of the formation shocks, such as Mach numbers, the turbulent spectra of the magnetic field, and the energy densities of shock-accelerated electrons. By suitably combining different cluster data, including Faraday rotation measures, we are able to measure the underlying properties of the plasma at the structure formation shocks and probe thermal properties of the warm-hot intergalactic medium. With enough radio sensitivity, shocks in the filamentary structure of the web should be illuminated as well as those in the cluster nodes we concentrate on here.