Understanding the mechanical properties of metal-organic frameworks (MOFs) is crucial not only to yield robust practical applications, but also to advance fundamental research underpinning the flexibility of a myriad of open-framework chemical compounds. Herein we present one of the most comprehensive structural analyses yet on MOF-mechanics: elucidating the complex elastic response of an isoreticular series of topical Zr-based MOFs, explaining all the important mechanical properties, and identifying major trends arising from systematic organic linker exchange. Ab initio density functional theory (DFT) was employed to establish the single-crystal elastic constants of the nanoporous MIL-140(A-D) structures, generating a complete 3-D representation of the principal mechanical properties, encompassing the Young's modulus, shear modulus, linear compressibility and Poisson's ratio. Of particular interest, we discovered significantly high values of both positive and negative linear compressibility and Poisson's ratio, whose framework molecular mechanisms responsible for such elastic anomalies have been fully revealed. In addition to pinpointing large elastic anisotropy and unusual physical properties, we analyzed the bulk modulus of isoreticular Zr-MOF compounds to understand the framework structural resistance against the hydrostatic pressure, and determined the averaged mechanical behaviour of bulk (polycrystalline) MOF materials important for the design of emergent applications.