In the theory of operads we consider generalized symmetric power functors defined by sums of coinvariant modules. One observes classically that the symmetric functor construction provides an isomorphism from the category of symmetric modules to a split subcategory of the category of functors on dg-modules (if dg-modules form our ground category). The purpose of this article is to obtain a similar relationship for functors on a category of algebras over an operad. Precisely, we observe that right modules over operads, symmetric modules equipped with a right operad action, give rise to functors on algebra categories and we prove that this construction yields a split embedding of categories. Then we check that right modules over operads form a model category. In addition we prove that the symmetric functor construction maps weak-equivalences of right modules to pointwise weak-equivalences of functors. As a conclusion, we obtain that right modules over operads supply good models for the homotopy of associated functors on algebra categories.