The mechanical properties play a vital role in the stability and behavior of clathrate hydrate. In this work, the structural and mechanical properties of some nitride gas (NH3, NO and N2O) hydrates were investigated using density functional theory calculations. The equilibrium lattice structures for these hydrates were obtained. The full second-order elastic constants were then determined by energy–strain analyses and the polycrystalline elastic properties were also predicted. It is found that these three gas hydrates have high elastic isotropy, but their shear properties are significantly different. This study may lay a theoretical foundation for future research on the structural evolution of clathrate hydrates under a mechanical field.