  Thermoelectric technology is an alternate way to efficiently utilize the
energy by converting waste heat into electricity. Thermoelectric requires
material with low thermal conductivity to improves its thermoelectric
performance. In this work, by solving Boltzmann transport equation based on
first principles calculations, we report an ultra-low room temperature thermal
conductivity of 2.08 Wm$^{-1}$K$^{-1}$ and 2.9 Wm$^{-1}$K$^{-1}$ along c-axis
and a-axis respectively for pure MgPb$_2$Te. To explain this ultra-low thermal
conductivity, we analyzed the elastic constants, phonon group velocity,
phonon-phonon scattering and contribution from transverse acoustic,
longitudinal acoustic and optical phonon branches. We also report the thermal
conductivity of MgPb$_2$Te nanostructures. At 50 nm, the room temperature
thermal conductivity of MgPb$_2$Te is 0.957 Wm$^{-1}$K$^{-1}$ and 1.459
Wm$^{-1}$K$^{-1}$ along c-axis and a-axis respectively. Ultra-low thermal
conductivity unraveled in this work shows MgPb$_2$Te would a promising material
for thermoelectric applications.
