We have studied the exciton spin relaxation times in In(x)Ga(1-x)N/GaN multiquantum wells as a function of well width and indium concentration for temperatures from 10 to 180 K. Well widths from 2 to 8 nm and indium concentrations from x = 0.02 to 0.15 have been investigated. In contrast to 1 nm wide quantum wells where spin beats were observed [J. Brown et al., Phys. Status Solidi B 243, 1643 (2006)], no spin beats were observed in any of our samples due to the fast spin relaxation times and a reduction in the exchange energy. In all samples for which a net spin polarization could be generated, the measured spin relaxation time was 1 ps or faster. The fast exciton spin decay time is caused by the influence of the holes via the exchange interaction, while the temperature dependence can be largely attributed to exciton-phonon scattering. In the widest wells (8 nm thick), the quantum confined Stark effect precluded the possibility of observing the spin dynamics. Similar measurements on an In(0.1)Ga(0.9)N epilayer yielded a spin relaxation time of 0.45 ps.