We present new measurements of the power spectra of the cosmic infrared background (CIB) anisotropies using the Planck 2015 full-mission HFI data at 353, 545, and 857 GHz over 20 000 square degrees. Unlike previous Planck measurements of the CIB power spectra, we do not rely on external HI data to remove Galactic dust emission from the Planck maps. Instead, we model the Galactic emission at the level of the power spectra, using templates constructed directly from the Planck data by exploiting the statistical isotropy of all extragalactic emission components. This allows us to work at the full resolution of Planck over large sky areas. We construct a likelihood based on the measured spectra (for multipoles 50 ≤ ℓ ≤ 2500) using analytic covariance matrices that account for masking and the realistic instrumental noise properties. The results of an MCMC exploration of this likelihood are presented, based on simple parameterised models of the CIB power that arises from clustering of infrared galaxies. We explore simultaneously the parameters describing the clustered power, the Poisson power levels, and the amplitudes of the Galactic power spectrum templates across the six frequency (cross-)spectra. The best-fit model provides a good fit to all spectra. As an example, Fig. 1 compares the measured auto spectra at 353, 545, and 857 GHz over 40% of the sky to the power in the best-fit model. We find that the power in the CIB anisotropies from galaxy clustering is roughly equal to the Poisson power at multipoles ℓ =2000 (the clustered power dominates on larger scales), and that our dust-cleaned CIB spectra are in good agreement with previous Planck and Herschel measurements. A key feature of our analysis is that it allows one to make many internal consistency tests. We show that our results are stable to data selection and choice of survey area, demonstrating both our ability to remove Galactic dust power to high accuracy and the statistical isotropy of the CIB signal.