We extract cosmological information from the anisotropic power-spectrum measurements from the recently completed Baryon Oscillation Spectroscopic Survey (BOSS), extending the concept of clustering wedges to Fourier space. Making use of new fast-Fourier-transform-based estimators, we measure the power-spectrum clustering wedges of the BOSS sample by filtering out the information of Legendre multipoles ℓ > 4. Our modelling of these measurements is based on novel approaches to describe non-linear evolution, bias and redshift-space distortions, which we test using synthetic catalogues based on large-volume N-body simulations. We are able to include smaller scales than in previous analyses, resulting in tighter cosmological constraints. Using three overlapping redshift bins, we measure the angular-diameter distance, the Hubble parameter and the cosmic growth rate, and explore the cosmological implications of our full-shape clustering measurements in combination with cosmic microwave background and Type Ia supernova data. Assuming a Λ cold dark matter (ΛCDM) cosmology, we constrain the matter density to Ω M= 0.311_{-0.010}^{+0.009} and the Hubble parameter to H_0 = 67.6_{-0.6}^{+0.7} km s^{-1 Mpc^{-1}}, at a confidence level of 68 per cent. We also allow for non-standard dark energy models and modifications of the growth rate, finding good agreement with the ΛCDM paradigm. For example, we constrain the equation-of-state parameter to w = -1.019_{-0.039}^{+0.048}. This paper is part of a set that analyses the final galaxy-clustering data set from BOSS. The measurements and likelihoods presented here are combined with others in Alam et al. to produce the final cosmological constraints from BOSS.