We present an analysis of high-resolution spectroscopy of several bright T Tauri stars using the VLT/CRIRES and Keck/NIRSPEC spectrographs, revealing the first detections of emission from HCN and C2H2 in circumstellar disks at near-infrared wavelengths. Using advanced data reduction techniques we achieve a dynamic range with respect to the disk continuum of ~500 at 3 microns, revealing multiple emission features of H2O, OH, HCN, and C2H2. We also present stringent upper limits for two other molecules thought to be abundant in the inner disk, CH4 and NH3. Line profiles for the different detected molecules are broad but centrally peaked in most cases, even for disks with previously determined inclinations of greater than 20 degrees, suggesting that the emission has both a Keplerian and non-Keplerian component as observed previously for CO emission. We apply two different modeling strategies to constrain the molecular abundances and temperatures: we use a simplified single-temperature LTE slab model with a Gaussian line profile to make line identifications and determine a best-fit temperature and initial abundance ratios, and we compare these values with constraints derived from a detailed disk radiative transfer model assuming LTE excitation but utilizing a realistic temperature and density structure. Abundance ratios from both sets of models are consistent with each other and consistent with expected values from theoretical chemical models, and analysis of the line shapes suggests the molecular emission originates from within a narrow region in the inner disk (R < 1 AU).