MV2Ga4 (M = Sc, Zr, Hf) compounds belong to an emerging class of materials showing a unique combination of unusual superconducting behavior with extended linear chains in the crystal structure. In order to gain insights into its mechanical and thermal properties, we have performed first-principles electronic-structure calculations in the framework of the Density Functional Theory (DFT). From the calculated second-order elastic constants, we have systematically shown that the extended linear vanadium chain substructures indeed give rise to an anisotropic regime in the elastic and mechanical moduli. The high density of valence and conduction electrons along the linear vanadium chains leads to a directional dependence of the reciprocal linear compressibility, Young’s modulus and shear modulus. Poisson’s ratio for several elongation directions is also drastically affected by the presence of extended V chains. If the elongation is along the V chains, all compounds exhibit practically the same Poisson ratio in directions perpendicular to it, further highlighting the importance of the V chains to the mechanical properties. Moreover, based on our results, we have discussed the possible consequences of the elastic anisotropy on the superconducting properties of the compounds. Finally, using the Debye–Grüneisen approximation, our calculations of thermal properties show a good agreement with the available experimental low temperature heat capacity data above the superconducting critical temperature.