We present the first results of a spectroscopic survey of Lyman break galaxies (LBGs) in the near-infrared aimed at detecting the emission lines of [O II], [O III], and Hβ from the H II regions of normal star-forming galaxies at z 3. From observations of 15 objects with the Keck telescope and the Very Large Telescope augmented with data from the literature for an additional four objects, we reach the following main conclusions. The rest-frame optical properties of LBGs at the bright end of the luminosity function are remarkably uniform, their spectra are dominated by emission lines, [O III] is always stronger than Hβ and [O II], and projected velocity dispersions are between 50 and 115 km s-1. Contrary to expectations, the star formation rates deduced from the Hβ luminosity are on average no larger than those implied by the stellar continuum at 1500 A; presumably any differential extinction between rest-frame optical and UV wavelengths is small compared to the relative uncertainties in the calibrations of these two star formation tracers. For the galaxies in our sample, the abundance of oxygen can only be determined to within 1 order of magnitude without recourse to other emission lines ([N II] and Hα), which are generally not available. Even so, it seems well established that LBGs are the most metal-enriched structures at z 3, apart from quasi-stellar objects, with abundances greater than about 1/10 solar and generally higher than those of damped Lyα systems at the same epoch. They are also significantly overluminous for their metallicities; this is probably an indication that their mass-to-light ratios are low compared to present-day galaxies. At face value, the measured velocity dispersions imply virial masses of about 1010 M☉ within half-light radii of 2.5 kpc. The corresponding mass-to-light ratios, M/L ≈ 0.15 in solar units, are indicative of stellar populations with ages between 108 and 109 yr, consistent with the UV-optical spectral energy distributions. However, we are unable to establish conclusively whether or not the widths of the emission lines reflect the motions of the H II regions within the gravitational potential of the galaxies, even though in two cases we see hints of rotation curves. All 19 LBGs observed show evidence for galactic-scale superwinds; such outflows have important consequences for regulating star formation, distributing metals over large volumes, and allowing Lyman continuum photons to escape and ionize the intergalactic medium.