Inelastic neutron and x-ray scattering measurements of phonons and spin waves in CuCrO2 were performed over a wide range of temperature, and complemented with first-principles simulations. The phonon dispersions and density of states are well reproduced by our density functional cal- culations, and reveal a strong anisotropy of Cu vibrations, with large amplitudes of low-frequency in-plane motions. In addition, we find that spin fluctuations persist above 300 K, far above the N eel temperature for long-range antiferromagnetic order, TN. Modeling of the thermal conductivity, based on our phonon measurements and simulations, reveals a significant anisotropy and indicates that the spin fluctuations above TN constitute a strong source of phonon scattering.