Several graph properties are characterized as a class of graphs that admit an orientation avoiding a finite set of oriented structures. For instance, if Fk is the set of homomorphic images of the directed path on k+1 vertices, then a graph is k-colourable if and only if it admits an orientation with no induced oriented subgraph in Fk. There is a fundamental question underlying this kind of characterizations: Given a graph property P, is there a finite set of oriented graphs F such that a graph belongs to P if and only if it admits an orientation with no induced oriented subgraph in F? We address this question by exhibiting some necessary conditions upon certain graph classes for them to admit such a characterization. Consequently, we exhibit an uncountable family of hereditary classes for which no such finite set exists. In particular, the class of graphs with no holes of prime length belongs to this family.