  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 FFT-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 CMB
and SN Ia data. Assuming a {\Lambda}CDM cosmology, we constrain the matter
density to {\Omega}_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 (CL) of 68 per cent. We
also allow for non-standard dark energy models and modifications of the growth
rate, finding good agreement with the {\Lambda}CDM 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 dataset from
BOSS. The measurements and likelihoods presented here are combined with others
in Alam et al. 2016 to produce the final cosmological constraints from BOSS.
