Recently introduced privacy legislation has aimed to restrict and control the amount of personal data published by companies and shared to third parties. Much of this real data is not only sensitive requiring anonymization, but also contains characteristic details from a variety of individuals. This diversity is desirable in many applications ranging from Web search to drug and product development. Unfortunately, data anonymization techniques have largely ignored diversity in its published result. This inadvertently propagates underlying bias in subsequent data analysis. We study the problem of finding a diverse anonymized data instance where diversity is measured via a set of diversity constraints. We formalize diversity constraints and study their foundations such as implication and satisfiability. We show that determining the existence of a diverse, anonymized instance can be done in PTIME, and we present a clustering-based algorithm. We conduct extensive experiments using real and synthetic data showing the effectiveness of our techniques, and improvement over existing baselines. Our work aligns with recent trends towards responsible data science by coupling diversity with privacy-preserving data publishing.