Various constrained frequent pattern mining problem formulations and associated algorithms have been developed that enable the user to specify various itemset-based constraints that better capture the underlying application requirements and characteristics. In this paper we introduce a new class of block constraints that determine the significance of an itemset pattern by considering the dense block that is formed by the pattern's items and its associated set of transactions. Block constraints provide a natural framework by which a number of important problems can be specified and make it possible to solve numerous problems on binary and real-valued datasets. However, developing computationally efficient algorithms to find these block constraints poses a number of challenges as unlike the different itemset-based constraints studied earlier, these block constraints are tough as they are neither anti-monotone, monotone, nor convertible. To overcome this problem, we introduce a new class of pruning methods that significantly reduce the overall search space and present a computationally efficient and scalable algorithm called CBMINER to find the closed itemsets that satisfy the block constraints.