Metal–organic frameworks (MOFs) need a high mechanical stability to be robust for industrial applications, like gas storage and catalysis. Their adsorption properties also correlate with the structure flexibility, which leads to the so-called breathing phenomenon. Finding relations between geometrical and topological descriptors and mechanical properties is important for explaining and predicting mechanical behavior of both synthesized and hypothetical MOFs. To address this, we present the full tensor DFT analysis of the second-order elastic constants for 22 either rigid or flexible MOFs assembled from rod secondary building units (rod MOFs) and related to 7 topological types of underlying nets: 4/5/t1, bik, crb, dia, gis, cda, and mog. The calculated values of the Young’s and shear moduli, linear compressibility, and Poisson’s ratio have found a good agreement with experimental observations. We have shown that the geometrical–topological features of the coordination framework predetermine the general form of the elastic tensor, while variations in the MOF composition tune the mechanical properties in specific directions. We report for the first time a negative linear compressibility for the CAU-10-OCH₃, NOTT-401, MIL-60, MIL-116, and MIL-118 frameworks. We have also revealed that the breathing behavior of six rod MOFs of the MIL-53, MIL-118, and CAU-10 families is caused by compliant geometrical–topological patterns and ligand–ligand interactions. The proposed classification of the geometrical–topological patterns into compliant and noncompliant can be used to search for and to design breathing MOFs.