Agricultural production residues are an easily accessible raw material for energy recovery in a circular economy. Therefore, the possibility of biogas production from herb processing waste, namely common thyme (Thymus vulgaris L.), peppermint (Mentha × piperita L.), curled mint (Mentha crispa L.), and currants (woody stems and leaves), was investigated. In this study, the evaluation of the natural biodegradability of plant waste under conditions typical for an agricultural biogas plant was consciously carried out without the application of pre-treatment processes (shredding, steam hydrolysis, chemical treatment) to facilitate the methane fermentation process. The average values of biogas production efficiency ranged from 75 to 320 m3/mg DM for herb species and from 152 to 209 m3/mg DM for currant varieties under normal conditions. As part of laboratory tests, the elemental composition, i.e., C, H, N, S, O, was determined. Moreover, the analysis showed the energy potential of the tested waste in thermochemical processes (combustion). Garden thyme residues have particularly high energy potential, as indicated by the high calorific value, low nitrogen and sulfur content, and low ash content.