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The metallic quasi-1D materials TaSe₃ and ZrTe₃ were demonstrated to have high current carrying capacity and resilience to scaling. These crystals consist of van der Waals planes of strongly bonded one-dimensional chains more weakly bonded to neighboring chains. To understand their thermal transport properties, we determined their phonon spectra and thermal conductivities using density functional theory and the phonon Boltzmann transport equation. The anisotropy of both the LA phonon velocities and thermal conductivities is larger in TaSe₃ than in ZrTe₃. The maximum LA phonon velocity in ZrTe₃ occurs in the cross-chain direction, which is a result of the strong cross-chain bonding. The maximum thermal conductivities for both materials occur in the chain directions. A significant percentage of the heat is carried by optical phonons. The phonon lifetimes and mean free paths for the low frequency modes is much shorter in TaSe₃ than in ZrTe₃ due to the presence of low-frequency optical modes and zone-folding features in TaSe₃. Thermal conductivities are calculated for different crystallographic directions over a range of temperatures [arXiv:2010.08165].