  We investigate the anisotropic clustering of the Baryon Oscillation
Spectroscopic Survey (BOSS) Data Release 12 (DR12) sample, which consists of
$1\,198\,006$ galaxies in the redshift range $0.2 < z < 0.75$ and a sky
coverage of $10\,252\,$deg$^2$. We analyse this dataset in Fourier space, using
the power spectrum multipoles to measure Redshift-Space Distortions (RSD)
simultaneously with the Alcock-Paczynski (AP) effect and the Baryon Acoustic
Oscillation (BAO) scale. We include the power spectrum monopole, quadrupole and
hexadecapole in our analysis and compare our measurements with a perturbation
theory based model, while properly accounting for the survey window function.
To evaluate the reliability of our analysis pipeline we participate in a mock
challenge, which resulted in systematic uncertainties significantly smaller
than the statistical uncertainties. While the high-redshift constraint on
$f\sigma_8$ at $z_{\rm eff}=0.61$ indicates a small ($\sim 1.4\sigma$)
deviation from the prediction of the Planck $\Lambda$CDM model, the
low-redshift constraint is in good agreement with Planck $\Lambda$CDM. 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~\citet{Alam2016} to produce the final cosmological constraints from BOSS.
