Metal-rich, mixed copper-rhodium sulfide Cu_3-δRh₃₄S₃₀ that represents a new Cu-filled variant of the Rh₁₇S₁₅ structure has been synthesized and structurally characterized. Copper content in the [CuRh₈] cubic cluster was found to vary notably dependent on the chosen synthetic route. Full site occupancy was achieved only in nanoscaled Cu₃Rh₃₄S₃₀ obtained by a rapid, microwave-assisted reaction of CuCl, Rh₂(CH₃CO₂)₄ and thiosemicarbazide at 300 °C in just 30 min; whereas merely Cu-deficient Cu_3-δRh₃₄S₃₀ (2.0 ≥ δ ≥ 0.9) compositions were realized via conventional high-temperature ceramic synthesis from the elements at 950 °C. Although Cu_3-δRh₃₄S₃₀ is metallic just like Rh₁₇S₁₅, the slightly enhanced metal content has a dramatic effect on the electronic properties. Whereas the Rh₁₇S₁₅ host undergoes a superconducting transition at 5.4 K, no signs of the latter were found for the Cu-derivatives at least down to 1.8 K. This finding is corroborated by the strongly reduced density of states at the Fermi level of the ternary sulfide and the disruption of long-range Rh-Rh interactions in favor of Cu-Rh interactions as revealed by quantum-chemical calculations.