Recent work on four dimensional effective descriptions of the heterotic string has identified the moduli of such systems as being given by kernels of maps between ordinary Dolbeault cohomology groups. The maps involved are defined by the supergravity data of the background solutions. Such structure is seen both in the case of Calabi-Yau compactifications with non-trivial constraints on moduli arising from the gauge bundle and in the case of some non-Kahler compactifications of the theory. This description of the moduli has allowed the explicit computation of the moduli stabilization effects of a wide range of non-trivial gauge bundles on Calabi-Yau three-folds. In this paper we examine to what extent the ideas and techniques used in this work can be extended to the case of flux compactifications of Type IIB string theory. Certain simplifications arise in the Type IIB case in comparison to the heterotic situation. However, complications also arise due to the richer supergravity data of the theory inducing a more involved map structure. We illustrate our discussion with several concrete examples of compactification of Type IIB string theory on conformal CICY three-folds with flux.