This article presents and discusses evidence that genitive and dative objects regularly become nominative in Ancient Greek passives of monotransitives and ditransitives. This is a typologically and theoretically significant state of affairs for two reasons. (i) As is well known, nonaccusative objects are, in many languages, not allowed to enter into Case alternations, a fact that has been accounted for in the government-binding/principles-and-parameters literature on the basis of the assumption that nonaccusative objects—prototypically datives—bear INHERENT, LEXICAL , or QUIRKY Case. By this reasoning, Ancient Greek genitives and datives must be concluded to have STRUCTURAL Case. (ii) Even in languages where dative-nominative (DAT-NOM) alternations do obtain, they are often limited to ditransitives, a fact that can been taken to suggest that dative qualifies as structural Case only in ditransitives. A language like Ancient Greek, which allows genitive and dative objects to become nominative in all passives (monotransitives and ditransitives), shows that it is, in principle, possible to have a linguistic system where genitive and dative qualify as structural Cases in both monotransitives and ditransitives. Case theories must be designed in such a way as to allow for this option. We argue for an analysis of Case alternations that combines the view that alternating datives and genitives enter the formal operation Agree with a morphological case approach to the distribution of overt case morphology. We furthermore compare Ancient Greek DAT-NOM and genitive-nominative (GEN-NOM) alternations in passives to Icelandic DAT-NOM and GEN-NOM alternations in middles, pointing to a number of interesting differences in the two types of alternations that depend on (i) the types of nonaccusative arguments entering Agree, (ii) the verbal head (Voice or v) entering Agree with nonaccusative objects, and (iii) the rules of dependent case assignment in connection to the role of nominative in the two languages.