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We study adjustable distributionally robust optimization problems, where their ambiguity sets can potentially encompass an infinite number of expectation constraints. Although such ambiguity sets have great modeling flexibility in characterizing uncertain probability distributions, the corresponding adjustable problems remain computationally intractable and challenging. To overcome this issue, we propose a greedy improvement procedure that consists of solving, via the (extended) linear decision rule approximation, a sequence of tractable subproblems—each of which considers a relaxed and finitely constrained ambiguity set that can be iteratively tightened to the infinitely constrained one. Through three numerical studies of adjustable distributionally robust optimization models, we show that our approach can yield improved solutions in a systematic way for both two-stage and multistage problems. History: Accepted by Pascal Van Hentenryck, Area Editor for Computational Modeling: Methods & Analysis. Funding: Financial support by the Early Career Scheme from the Hong Kong Research Grants Council [Project No. CityU 21502820], the CityU Start-Up Grant [Project No. 9610481], the CityU Strategic Research Grant [Project No. 7005688], the National Natural Science Foundation of China [Project No. 72032005], and Chow Sang Sang Group Research Fund sponsored by Chow Sang Sang Holdings International Limited [Project No. 9229076] is gratefully acknowledged. Supplemental Material: The software that supports the findings of this study is available within the paper and its Supplemental Information (https://pubsonline.informs.org/doi/suppl/10.1287/ijoc.2021.0181), as well as from the IJOC GitHub software repository (https://github.com/INFORMSJoC/2021.0181). The complete IJOC Software and Data Repository is available at https://informsjoc.github.io/. [ABSTRACT FROM AUTHOR]