Molybdenum ditelluride (MoTe) is an unique transition metal dichalcogenide owing to its energetically comparable 1H and 1T′ phases. This implies a high chance of coexistence of 1H/1T′ heterostructures which poses great complexity in the measurement of the intrinsic lattice thermal conductivities (κ). In this work, via first-principles calculations, we examine the lattice-wave propagation and thermal conduction in this highly structurally anisotropic 1T′ MoTe. Our calculation shows that the 1T′ phase has a sound velocity of 2.13 km/s (longitudinal acoustic wave), much lower than that of the 1H phase (4.05 km/s), indicating a staggered transmission of lattice waves across the boundary from 1H to 1T′ phase. Interestingly, the highly anisotropic 1T′ MoTe shows nearly isotropic and limited κ of 13.02 W/mK, owing to a large Grüneisen parameter of acoustic flexural mode, heavy masses of Mo and Te elements and a low phonon group velocity. Accumulative κ as a function of mean free path (MFP) indicates phonons with MFP less than ∼ 300 nm contribute 80 % of κ and an inflection point at ∼ 600 nm. Our results will be critical for understanding of the size dependent κ of nanostructured 1T′ MoTe.