We present new wideband radio observations with the Australia Telescope Compact Array of a sample of 12 giant radio galaxies. The radio observations are part of a larger radio-optical study aimed at relating the radio structures with the ambient medium on large scales. With projected linear sizes larger than 0.7 Mpc, these objects are ideal candidates for the study of the warm-hot intergalactic medium (WHIM). The sample includes sources with sizes spanning 0.8-3.2 Mpc and total powers of 1.2 × 1024 to 4.0 × 1026 WHz-1 at 2.1 GHz. Redshifts were limited to z≤0.15 to permit spectroscopic observations of the hosts and neighbouring galaxies, which were obtained using the AAOmega spectrograph on the Anglo-Australian Telescope. We derive lobe energy densities from the radio observations via equipartition arguments. The inferred pressures in the lobes of the giant radio sources,which range from 1.1×10-15 to 2.0× 10-14 Pa (80 to 1500 cm-3 K), are lower than previously inferred from X-ray observations of dense filaments. Comparison with the OverWhelmingly Large Simulations suggests that the WHIM in pressure balance with the radio lobes has a temperature in excess of ~106.5 K or a particle overdensity in the range 50-500. This study highlights the capability of next generation surveys, such as the Evolutionary Map of the Universe survey with the Australian Square Kilometre Array Pathfinder, to study populations of giant radio sources at lower surface brightness and thereby discriminate between models for the cosmological evolution of the intergalactic medium and examine the validity of cosmological hydrodynamical simulations.