K-Anonymity has been proposed as a mechanism for protecting privacy in microdata publishing, and numerous recoding "models" have been considered for achieving 𝑘anonymity. This paper proposes a new multidimensional model, which provides an additional degree of flexibility not seen in previous (single-dimensional) approaches. Often this flexibility leads to higher-quality anonymizations, as measured both by general-purpose metrics and more specific notions of query answerability. Optimal multidimensional anonymization is NP-hard (like previous optimal 𝑘-anonymity problems). However, we introduce a simple greedy approximation algorithm, and experimental results show that this greedy algorithm frequently leads to more desirable anonymizations than exhaustive optimal algorithms for two single-dimensional models.