The CuI-based delafossite structure, CuIMIIIO2, can accommodate a wide range of rare earth and transition metal cations on the MIII site. Substitutional doping of divalent ions for these trivalent metals is known to produce higher p-type conductivity than that occurring in the undoped materials. However, an explanation of the conductivity anomalies observed in these p-type materials, as the trivalent metal is varied, is still lacking. In this article, we examine the electronic structure of CuIMIIIO2 (MIII=Al,Cr,Sc,Y) using density functional theory corrected for on-site Coulomb interactions in strongly correlated systems (GGA+U) and discuss the unusual experimental trends. The importance of covalent interactions between the MIII cation and oxygen for improving conductivity in the delafossite structure is highlighted, with the covalency trends found to perfectly match the conductivity trends. We also show that calculating the natural band offsets and the effective masses of the valence band maxima is not an ideal method to classify the conductionproperties of these ternary materials.