We calcuate lattice constants, volumes, elastic moduli, Vickers hardnesses and density of states of the tetragonal, monoclinic and orthorhombic Ge₃N₄.Fundamental properties obtained are compared with experimental data and other theoretical results, which shows good agreement.Negative formation enthalpy, linear relation between lattice deformation and pressure, and volume show that three phases retain stabilities in a pressure range of 0~20 GPa.Density of states show that Ge₃N₄ are semiconductors with strong s-p hybridizations.All polymorphs are brittle materials but anisotropic.Several thermodynamic properties are investigated with quasi-harmonic approach considering lattice vibrations and anharmonic effects.Helmholz free energy and internal energy show strong dependences on pressure.Interesting features are seen in heat capacity of the novel Ge₃N₄ nitrides, especially at high temperatures.