Using $1/S$ expansion, we discuss the magnon spectrum of Heisenberg antiferromagnet (AF) on a simple cubic lattice with small dipolar interaction at small temperature $Tll T_N$, where $T_N$ is the Neel temperature. Similar to 3D and 2D ferromagnets,
quantum and thermal fluctuations renormalize greatly the bare gapless spectrum leading to a gap $Deltasim omega_0$, where $omega_0$ is the characteristic dipolar energy. This gap is accompanied by anisotropic corrections to the free energy which make the cube edges easy directions for the staggered magnetization (dipolar anisotropy). In accordance with previous results, we find that dipolar forces split the magnon spectrum into two branches. This splitting makes possible two types of processes which lead to a considerable enhance of the damping compared to the Heisenberg AF: a magnon decay into two other magnons and a confluence of two magnons. It is found that magnons are well defined quasiparticles in quantum AF. We demonstrate however that a small fraction of long-wavelength magnons can be overdamped in AFs with $Sgg1$ and in quantum AFs with a single-ion anisotropy competing with the dipolar anisotropy. Particular materials are pointed out which can be suitable for experimental observation of this long-wavelength magnons breakdown that contradicts expectation of the quasiparticle concept.
