We have measured the conductance of single-molecule junctions created with three different molecular wires using the scanning tunneling microscope-based break-junction technique. Each wire contains one of three different cyclic five-membered rings: cyclopentadiene, furan, or thiophene. We find that the single-molecule conductance of these three wires correlates negatively with the resonance energy of the five-membered ring; the nonaromatic cyclopentadiene derivative has the highest conductance, while the most aromatic of this series, thiophene, has the lowest. Furthermore, we show for another wire structure that the conductance of furan-based wires is consistently higher than for analogous thiophene systems, indicating that the negative correlation between conductance and aromaticity is robust. The best conductance would be for a quinoid structure that diminishes aromaticity. The energy penalty for partly adopting the quinoid structure is less with compounds having lower initial aromatic stabilization. An additional effect may reflect the lower HOMOs of aromatic compounds.