The nearby brown dwarf LP 944-20 has been monitored for radial velocity variability at optical and near-infrared wavelengths using the VLT/UVES and the Keck/NIRSPEC, respectively. The UVES radial velocity data obtained over 14 nights spanning a baseline of 841 days show significant variability with an amplitude of 3.5 km s-1. The periodogram analysis of the UVES data indicates a possible period between 2.5 and 3.7 hr, which is likely due to the rotation of the brown dwarf. However, the NIRSPEC data obtained over 6 nights show an rms dispersion of only 0.36 km s-1 and do not follow the periodic trend. These results indicate that the variability seen with UVES is likely to be due to rotationally modulated inhomogeneous surface features. We suggest that future planet searches around very low mass stars and brown dwarfs using radial velocities will be better conducted in the near-infrared than in the optical.