High signal-to-noise ratio, low-resolution spectra have been obtained for 22 globular clusters (GCs) in NGC 4365. Some of these were selected as probable representatives of an intermediate-age (2-5 Gyr), extremely metal-rich GC subpopulation. The presence of such a subpopulation had been inferred from the unusual optical and near-infrared color distributions of GCs in this otherwise typical Virgo elliptical galaxy. However, ages derived from Lick indices are consistent with uniformly old mean ages for all GCs in our sample. The metallicities of the clusters show some evidence of a trimodal distribution. The most metal-poor and metal-rich peaks are consistent with the values expected for an elliptical galaxy of this luminosity, but there appears to be an additional, intermediate-metallicity peak lying between them. New Hubble Space Telescope photometry is consistent with this result. A plausible scenario is that in earlier data these three peaks merged into a single broad distribution. Our results suggest that it is difficult to identify intermediate-age GC subpopulations solely with photometry, even when both optical and near-infrared colors are used.