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Weakly-supervised thoracic disease detection in chest X-rays (CXR) is challenging due to subtle appearances and complex anatomical overlap, motivating anatomy-aware modeling for improved localization. However, prior anatomy-aware methods typically rely on coarse region proxies or static spatial priors, which may restrict dynamic instance discovery and limit precise localization of small abnormalities. We propose Anatomy-Structured Hierarchical Multiple Instance Learning (ASH-MIL), a framework that introduces parallel anatomy-structured observation branches (cardiac, pulmonary, and agnostic) combined with hierarchical MIL aggregation. Anatomical priors are injected as soft spatial biases into decoder cross-attention, enabling anatomically grounded evidence maps without disease bounding-box supervision. Instance localization is derived directly from MIL-weighted cross-attention maps. Experiments on CXR8 and cross-domain MIMIC-CXR held-out sets demonstrate consistent improvements over prior weakly-supervised and anatomy-aware approaches, particularly under stricter localization criteria. Our code is available at https://github.com/jn-kim/ash-mil.