A group of triethylphosphine gold(I) and silver(I) complexes, structurally related to auranofin, were prepared and investigated as potential anticancer drug candidates. The antiproliferative properties of these metal compounds were assessed against two leukemia cell lines, i.e., CCRF-CEM and its multidrug-resistant counterpart, CEM/ADR5000. Interestingly, potent cytotoxic effects were disclosed for both series of compounds against leukemia cells, with IC50 values generally falling in the low-micromolar range, the gold derivatives being on the whole more effective than the silver analogues. Some initial structure-function relationships were drawn. Subsequently, the ability of the study compounds to inhibit the three main catalytic activities of the proteasome was investigated. Different patterns of enzyme inhibition emerged for the various metal complexes. Notably, gold compounds were able to inhibit effectively both the trypsin-like and chymotrypsin-like proteasome activities, being less effective toward the caspase-like catalytic activity. In most cases, a significant selectivity of the study compounds toward the proteasome proteolytic activities was detected when compared to other proteases. The implications of the obtained results are discussed.