We developed a modular framework to obtain exact and approximate solutions to a class of linear optimization problems with recourse with the goal to minimize the worst-case expected objective over an ambiguity set of distributions. The ambiguity set is specified by linear and conic quadratic representable expectation constraints and the support set is also linear and conic quadratic representable. We propose an approach to lift the original ambiguity set to an extended one by introducing additional auxiliary random variables. We show that by replacing the recourse decision functions with generalized linear decision rules that have affine dependency on the uncertain parameters and the auxiliary random variables, we can obtain good and sometimes tight approximations to a two-stage optimization problem. This approach extends to a multistage problem and improves upon existing variants of linear decision rules. We demonstrate the practicability of our framework by developing a new algebraic modeling package named ROC, a C++ library that implements the techniques developed in this paper.