[abridged] We present an anisotropic analysis of the baryonic acoustic oscillation (BAO) scale in the twelfth and final data release of the Baryonic Oscillation Spectroscopic Survey (BOSS). We independently analyse the LOWZ and CMASS galaxy samples: the LOWZ sample contains contains 361\,762 galaxies with an effective redshift of $z_{\rm LOWZ}=0.32$; the CMASS sample consists of 777\,202 galaxies with an effective redshift of $z_{\rm CMASS}=0.57$. We extract the BAO peak position from the monopole power spectrum moment, $\alpha_0$, and from the $\mu^2$ moment, $\alpha_2$, where $\mu$ is the cosine of the angle to the line-of-sight. The $\mu^2$-moment provides equivalent information to that available in the quadrupole but is simpler to analyse. After applying a reconstruction algorithm to reduce the BAO suppression by bulk motions, we measure the BAO peak position in the monopole and $\mu^2$-moment, which are related to radial and angular shifts in scale. We report $H(z_{\rm LOWZ})r_s(z_d)=(11.60\pm0.60)\cdot10^3\,{\rm km}s^{-1}$ and $D_A(z_{\rm LOWZ})/r_s(z_d)=6.66\pm0.16$ with a cross-correlation coefficient of $r_{HD_A}=0.41$, for the LOWZ sample; and $H(z_{\rm CMASS})r_s(z_d)=(14.56\pm0.37)\cdot10^3\,{\rm km}s^{-1}$ and $D_A(z_{\rm CMASS})/r_s(z_d)=9.42\pm0.13$ with a cross-correlation coefficient of $r_{HD_A}=0.47$, for the CMASS sample. We combine these results with the measurements of the BAO peak position in the monopole and quadrupole correlation function of the same dataset \citep[][companion paper]{Cuestaetal2015} and report the consensus values: $H(z_{\rm LOWZ})r_s(z_d)=(11.63\pm0.69)\cdot10^3\,{\rm km}s^{-1}$ and $D_A(z_{\rm LOWZ})/r_s(z_d)=6.67\pm0.15$ with $r_{HD_A}=0.35$ for the LOWZ sample; $H(z_{\rm CMASS})r_s(z_d)=(14.67\pm0.42)\cdot10^3\,{\rm km}s^{-1}$ and $D_A(z_{\rm CMASS})/r_s(z_d)=9.47\pm0.12$ with $r_{HD_A}=0.52$ for the CMASS sample.