By using state-of-the-art atomistic methods we provide an accurate estimate of thermal conductivity of methylammonium lead halide as a function of sample size and temperature, in agreement with experimental works. We show that the thermal conductivity of methylammonium lead halide is intrinsically low, due to the low sound velocity of the PbI lattice. Furthermore, by selectively analyzing the effect of different molecular degrees of freedom, we clarify the role of the molecular substructure by showing that the internal modes above 150 cm⁻¹ (in addition to rotations) are effective in reducing the thermal conductivity of hybrid perovskites. This analysis suggests strategies to tailor the thermal conductivity by modifying the internal structure of organic cations.