We present an x-ray microtomography study of the compaction process of cylindrical rods under tapping. The process is monitored by measuring the evolution of the orientational order parameter, local, and overall packing densities as a function of the tapping number for different tapping intensities. The slow relaxation dynamics of the orientational order parameter can be well fitted with a stretched-exponential law with stretching exponents ranging from 0.9 to 1.6. The corresponding relaxation time versus tapping intensity follows an Arrhenius behavior which is reminiscent of the slow dynamics in thermal glassy systems. We also investigated the boundary effect on the ordering process and found that boundary rods order faster than interior ones. In searching for the underlying mechanism of the slow dynamics, we estimated the initial random velocities of the rods under tapping and found that the ordering process is compatible with a diffusion mechanism. The average coordination number as a function of the tapping number at different tapping intensities has also been measured, which spans a range from 6 to 8.