The single crystal elastic constants, polycrystalline elastic moduli and related properties of orthorhombic MgSiN₂, MgGeN₂ and MgSnN₂ have been calculated using density functional theory and compared to the related wurtzite structured AlN, GaN and InN. Since there are no experimental studies of single crystal elastic properties of neither MgSiN₂, MgGeN₂ or MgSnN₂, we have established the accuracy of the calculations by comparison with experimental data for AlN, GaN and InN. The calculated polycrystalline elastic moduli of MgSiN₂ are found to be in good agreement with available experimental elastic moduli. It will be shown that MgSiN₂ and MgGeN₂ have a small xy-plane lattice mismatch with AlN and GaN, respectively, while at the same time being significantly softer than both AlN and GaN. This shows that MgSiN₂ and MgGeN₂ should be possible to be grown on AlN and GaN without significant lattice mismatch or strain.