While the transport of ions and electrons in conventional Li-ion battery cathode materials is well understood, our knowledge of the phonon (heat) transport is still in its infancy. We present a first-principles theoretical investigation of the chemical trends in the phonon frequency dispersion, mode lifetimes, and thermal conductivity in the series of layered lithium transition-metal oxides Li(NixMnyCoz)O2 (x + y + z = 1). The oxidation and spin states of the transition metal cations are found to strongly influence the structural dynamics. Calculations of the thermal conductivity show that LiCoO2 has highest average conductivity of 45.9 W·m-1·K-1 at T = 300 K and the largest anisotropy, followed by LiMnO2 with 8.9 W·m-1·K-1 and LiNiO2 with 6.0 W·m-1·K-1. The much lower thermal conductivity of LiMnO2 and LiNiO2 is found to be due to 1-2 orders of magnitude shorter phonon lifetimes. We further model the properties of binary and ternary transition metal combinations to examine the possible effects of mixing on the thermal transport. These results serve as a guide to ongoing work on the design of multicomponent battery electrodes with more effective thermal management.