Particle acceleration in magnetized relativistic jets still puzzles theorists, specially when one tries to explain the highly variable emission observed in blazar jets or gamma-ray bursts putting severe constraints on current models. In this work we investigate the acceleration of particles injected in a three-dimensional relativistic magnetohydrodynamical jet subject to current driven kink instability (CDKI), which drives turbulence and fast magnetic reconnection. We find that once the turbulence is fully developed in the jet, achieving a nearly stationary state, the amplitude of the excited wiggles along the jet spine also attains a maximum growth, causing the disruption of the magnetic field lines and the formation of several sites of fast reconnection. This occurs after the CDKI achieves a plateau in its non-linear growth. Test protons injected in the nearly stationary snapshots of the jet, experience an exponential acceleration up to a maximum energy. For a background magnetic field of B ∼ 0.1 G, this saturation energy is ∼ 10 16 eV, while for B ∼ 10 G it is ∼ 10 18 eV. The Larmor radius of the particles attaining the saturation energy corresponds to the size of the acceleration region, being of the order of the diameter of the perturbed jet. This regime of particle acceleration is very similar in all these evolved snapshots and lasts for several hundred hours until the saturation energy. The simulations also reveal a clear association of the accelerated particles with the regions of fast reconnection, indicating its dominant role on the acceleration process. Beyond those saturation values, the particles suffer further acceleration to energies up to 100 times larger, but at a slower rate due to drift in the varying field. In the early stages of the development of the non-linear growth of CDKI in the jet, when there are still no sites of fast reconnection, injected particles are also efficiently accelerated, but by magnetic curvature drift in the wiggling jet spine. However, in order to particles to be accelerated by this process, they have to be injected with an initial energy much larger than that required for particles to accelerate in reconnection sites. Finally, we have also obtained from the simulations an acceleration time due to reconnection with a weak dependence on the particles energy E, t A ∝ E 0.1 . The energy spectrum of the accelerated particles develops a high energy tail with a power law index p ∼ -1.2 in the beginning of the acceleration, in agreement with earlier works. Our results provide an appropriate multi-dimensional framework for exploring this process in real systems and explain their complex emission patterns,