Magnetization ($\mathbit{M}$) precessions driven by ultrafast laser-induced nonthermal effects are observed in undoped yttrium iron garnet (YIG) films of (100) and (111) orientations using pump-probe time-resolved magneto-optical Kerr effect. The $\mathbit{M}$ precessions show a strong dependence on the polarization direction of linearly polarized pump pulses of 400 nm. In contrast, we can barely observe any $\mathbit{M}$ precession using circularly polarized pump pulses, which indicates that the inverse Faraday effect is negligible. For the case of linear pump polarization, a phenomenological model is introduced, based on the modulation of $\mathbit{M}$ via a modulation of fourth-rank susceptibility tensors by a laser pulse. This allows one to distinguish the contributions of the inverse Cotton-Mouton effect (ICME) from those of the photoinduced magnetic anisotropy (PMA). Using the formula derived from the phenomenological model, we perform the fitting of the polarization-direction-dependent precession phase and amplitude in (100)- and (111)-oriented YIG films. The fitting results reveal that the $\mathbit{M}$-precession excitation originates from a combination effect of ICME and PMA, but the ICME plays the dominant role.