We numerically investigate the kinematic properties of globular cluster systems (GCSs) in E/S0 galaxies formed from dissipationless merging of spiral galaxies. The metal-poor globular clusters (MPCs) and metal-rich clusters (MRCs) in the merger progenitors are initially assumed to have spatial distributions consistent with the Milky Way GCS. Our principal results, which can be tested against observations, are as follows. Both MPCs and MRCs in elliptical galaxies formed from major mergers can exhibit significant rotation at large radii (~20 kpc) due to the conversion of initial orbital angular momentum into intrinsic angular momentum of the remnant. MPCs show higher central velocity dispersions than MRCs for most major merger models. Vm/σ0 (where Vm and σ0 are the GCS maximum rotational velocity and central velocity dispersion, respectively) ranges from 0.2 to 1.0 and from 0.1 to 0.9 for the MPCs and MRCs, respectively, within 6Re for the remnant elliptical. For most merger remnant ellipticals, Vm/σ0 of GCSs within 6Re is greater than that of the field stars within 2Re. The radial profiles of rotational velocities and velocity dispersions of the GCSs depend upon the orbital configuration of the merger progenitors, their mass ratios and the viewing angle. For example, more flattened early-type galaxies, formed through mergers with small mass ratios (~0.1), show little rotation in the outer MRCs. Two-dimensional (2D) velocity dispersion distributions of the GCSs of merger remnant ellipticals are generally flattened for both MPCs and MRCs, reflecting the fact that the GCSs have anisotropic velocity dispersions. The 2D distributions of the line-of-sight velocity of the GCSs in some remnant ellipticals show minor-axis rotation, particularly for MRCs. The kinematic properties of MPCs in merger remnant ellipticals strongly resemble those of the surrounding dark matter. This implies that the kinematics of MPCs in such galaxies can be used to probe the kinematic properties of their dark matter haloes. We discuss these results in the context of GC and galaxy formation. We note a possible difference in the GC kinematics between field and cluster Es and explain how GC kinematics may help us understand the origin of S0 galaxies.