Based on the Hartree-Fock approximation the author proposes a nonlinear spin-wave theory of anisotropic 3D (or quasi-2D) quantum Heisenberg antiferromagnets, which reduces to the known spin-wave theories of isotropy when the anisotropy parameter delta takes special values. In the Hartree-Fock approximation the Dyson-transformed Hamiltonian is equivalent to the Holstein-Primakoff-transformed Hamiltonian truncated up to quartic operator terms. The spin-wave lifetime is obtained in the first-order approximation. For very small delta , the Neel temperature T_N is much smaller than the coupling constant J, in contrast with T_N approximately J in the 3D isotropic case, so that the nonlinear anisotropic spin-wave theory is suitable for a description of the ordering phase as well as the paramagnetic phase (up to J) of layer-like antiferromagnets. Applied to the antiferromagnetism of the cuprate La₂CuO₄ the quasi-2D nonlinear spin-wave theory describes quite satisfactorily the existing experimental data of the Neel transition temperature, the correlation length above the Neel temperature, and staggered magnetization of the material if J=1034 K and the anisotropy parameter is set to be 4*10^-5.