We report the results of a systematic near-IR spectroscopic survey using the Subaru, VLT, and Keck Telescopes of a sample of high-redshift ULIRGs mainly composed of submillimeter-selected galaxies. Our observations span the rest-frame optical range containing nebular emission lines such as H, [O iii] kk4959, 5007, and [O ii] k3727, which are essential for making robust diagnostics of the physical properties. Using the H/H emission line ratios, we derive internal extinction estimates for these galaxies similar to those of local ULIRGs: A V $ 2:9 AE 0:5. Correcting the H estimates of the star formation rate for dust extinction results in rates that are consistent with those estimated from the far-IR luminosity. The majority (>60%) of our sample show spectral features characteristic of AGNs (although this partially reflects an observational bias), with $65% exhibiting broad Balmer emission lines. A proportion of these sources show relatively low [O iii] k5007 / H line ratios, which are similar to those of narrow-line Seyfert 1 galaxies, suggesting low-mass black holes that are rapidly growing. In the subsample of our survey with both [O iii] k5007 and hard X-ray coverage, at least $60% show an excess of [O iii] k5007 emission, by a factor of 5Y10, relative to the hard X-ray luminosity compared to the correlation between these two properties seen in Seyfert galaxies and QSOs locally. From our spectral diagnostics, we propose that the strong [O iii] k5007 emission in these galaxies arises from shocks in dense gaseous regions. Due to sensitivity and resolution limits, our sample is biased to strong-line emitters and hence our results do not yet provide a complete view of the physical properties of the whole high-redshift ULIRG population.