We use the eta invariant of Spin^c Dirac operators to distinguish connected components of moduli spaces of  Riemannian metrics with positive Ricci curvature.  We then find infinitely many non-diffeomorphic five manifolds \((#^k S^2 \times S^3)/\mathbb{Z}_2\) for which these moduli spaces each have infinitely many components.  The manifolds are total spaces of principal \(S^1\) bundles and the metrics are lifted from Ricci positive metrics on the bases using a method of Gilkey, Park, and Tuschmann.  We compute the eta invariants by extending metrics and auxiliary connections over the associated disc bundles, generalizing a technique of Kreck and Stolz.