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. 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 $B sim 0.1$ G, this saturation energy is $sim 10^{16}$ eV, while for $B sim 10$ G it is $sim 10^{18}$ eV. The simulations also reveal a clear association of the accelerated particles with the regions of fast reconnection. 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, 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 $E$, $t_A propto E^{0.1}$. The energy spectrum of the accelerated particles develops a high energy tail with a power law index $p sim$ -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, specially in the very high energy bands and the associated neutrino emission recently detected in some blazars.
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