We present new results on the Lyα emission-line kinematics of 18 z ~ 2-3 star-forming galaxies with multiple-peaked Lyα profiles. With our large spectroscopic database of UV-selected star-forming galaxies at these redshifts, we have determined that ~30% of such objects with detectable Lyα emission display multiple-peaked emission profiles. These profiles provide additional constraints on the escape of Lyα photons due to the rich velocity structure in the emergent line. Despite recent advances in modeling the escape of Lyα from star-forming galaxies at high redshifts, comparisons between models and data are often missing crucial observational information. Using Keck II NIRSPEC spectra of Hα (z ~ 2) and [O III]λ5007 (z ~ 3), we have measured accurate systemic redshifts, rest-frame optical nebular velocity dispersions, and emission-line fluxes for the objects in the sample. In addition, rest-frame UV luminosities and colors provide estimates of star formation rates and the degree of dust extinction. In concert with the profile sub-structure, these measurements provide critical constraints on the geometry and kinematics of interstellar gas in high-redshift galaxies. Accurate systemic redshifts allow us to translate the multiple-peaked Lyα profiles into velocity space, revealing that the majority (11/18) display double-peaked emission straddling the velocity-field zero point with stronger red-side emission. Interstellar absorption-line kinematics suggest the presence of large-scale outflows for the majority of objects in our sample, with an average measured interstellar absorption velocity offset of (Δv_(abs))=–230 km s^(–1). A comparison of the interstellar absorption kinematics for objects with multiple- and single-peaked Lyα profiles indicate that the multiple-peaked objects are characterized by significantly narrower absorption line widths. We compare our data with the predictions of simple models for outflowing and infalling gas distributions around high-redshift galaxies. While popular "shell" models provide a qualitative match with many of the observations of Lyα emission, we find that in detail there are important discrepancies between the models and data, as well as problems with applying the framework of an expanding thin shell of gas to explain high-redshift galaxy spectra. Our data highlight these inconsistencies, as well as illuminating critical elements for success in future models of outflow and infall in high-redshift galaxies