We measure and analyse the clustering of the Baryon Oscillation Spectroscopic Survey (BOSS) relative to the line of sight (LOS), for LOWZ and CMASS galaxy samples drawn from the final Data Release 12. The LOWZ sample contains 361 762 galaxies with an effective redshift of z lowz = 0.32, and the CMASS sample 777 202 galaxies with an effective redshift of z cmass = 0.57. From the power spectrum monopole and quadrupole moments around the LOS, we measure the growth of structure parameter f times the amplitude of dark matter density fluctuations σ8 by modelling the redshift-space distortion signal. When the geometrical Alcock–Paczynski effect is also constrained from the same data, we find joint constraints on fσ8, the product of the Hubble constant and the comoving sound horizon at the baryon-drag epoch H(z)r s(z d), and the angular distance parameter divided by the sound horizon D A(z)/r s(z d). We find f(z lowz)σ8(z lowz) = 0.394 ± 0.062, D A(z lowz)/r s(z d) = 6.35 ± 0.19, H(z lowz)r s(z d) = (11.41 ± 0.56) 103 km s− 1 for the LOWZ sample, and f(z cmass)σ8(z cmass) = 0.444 ± 0.038, D A(z cmass)/r s(z d) = 9.42 ± 0.15, H(z cmass)r s(z d) = (13.92 ± 0.44) 103 km s− 1 for the CMASS sample. We find general agreement with previous BOSS DR11 measurements. Assuming the Hubble parameter and angular distance parameter are fixed at fiducial Λcold dark matter values, we find f(z lowz)σ8(z lowz) = 0.485 ± 0.044 and f(z cmass)σ8(z cmass) = 0.436 ± 0.022 for the LOWZ and CMASS samples, respectively.