We show that terahertz plasmons in AlGaN/GaN grating-gate structures efficiently modulate the reflection of room temperature thermal radiation, leading to spectra that are in agreement with the measurements of plasmon absorption using high-power external sources. For typical samples of a few square millimeters in size, the reflected radiation intensity is relatively weak, and measurements need the use of gate voltage plasmon modulation and lock-in detection techniques. We show that unintentional use of lock-in techniques may lead to artifacts and demonstrate what kind of special precautions need to be taken into account. We show that drain voltage modulation also leads to modulation of the reflected thermal radiation by plasmons. Our results are of key importance for the research on new resonant plasmon-based terahertz sources because of the always present superposition of electrically excited terahertz emission and background radiation reflected from the structures.