We have obtained spectroscopic observations of the nuclear regions of 42 galaxies known to harbor strong OH masers. These megamaser galaxies represent a subsample of FIR (ultra)luminous galaxies, which typically have FIR luminosities in excess of 1011L☉. The primary goal of this study is to investigate the nuclear activity sources of OH megamaser galaxies. We are able to classify the nuclear emission-line spectra of all but one of our sample, and we find that this class of galaxies is dominated by active galactic nuclei (AGNs), although starburst galaxies do make up an appreciable fraction of the megamasers as well. Fully 45% of the megamasers exhibit Seyfert or LINER spectra, predominantly of Seyfert 2 type, although two galaxies with broad lines are observed. This observation is consistent with the current unification models for the two types of Seyfert activity, since the assumed geometry necessary to detect a megamaser places the Seyfert nucleus behind a high column density of molecular gas (i.e., the molecular torus). Starburst-nucleus galaxies comprise 32.5% of our sample, while 22.5% are classified as ``composite nuclear spectra'' (CSN) sources, showing evidence of both AGN and starburst activity. A number of objects show unusual emission-line ratios, not surprising for a group of galaxies that are known a priori to possess substantial absorbing material along the line of sight to their nuclei. Our results are compared to previous studies of FIR-selected galaxy samples, as well as to radio-continuum observations of these galaxies. The activity classification obtained from the radio data disagrees with the optical classifications in roughly 25% of the sources; we discuss possible explanations for these discrepancies.