We report the results from a systematic search for molecular (OH 119 μm) outflows with Herschel-PACS† in a sample of 43 nearby (z < 0.3) galaxy mergers, mostly ultraluminous infrared galaxies (ULIRGs) and QSOs. We find that the character of the OH feature (strength of the absorption relative to the emission) correlates with that of the 9.7-μm silicate feature, a measure of obscuration in ULIRGs. Unambiguous evidence for molecular outflows, based on the detection of OH absorption profiles with median velocities more blueshifted than −50 km s−1, is seen in 26 (70%) of the 37 OH-detected targets, suggesting a wide-angle (~ 145°) outflow geometry. Conversely, unambiguous evidence for molecular inflows, based on the detection of OH absorption profiles with median velocities more redshifted than +50 km s−1, is seen in only 4 objects, suggesting a planar or filamentary geometry for the inflowing gas. Terminal outflow velocities of ~ −1000 km s−1 are measured in several objects, but median outflow velocities are typically ~ −200 km s−1. While the outflow velocities show no statistically significant dependence on the star formation rate, they are distinctly more blueshifted among systems with large AGN fractions and luminosities [log (LAGN/L⊙) ≥ 11.8 ± 0.3]. The quasars in these systems play a dominant role in driving the molecular outflows. In contrast, the most AGN dominated systems, where OH is seen purely in emission, show relatively modest OH line widths, despite their large AGN luminosities, perhaps indicating that molecular outflows subside once the quasar has cleared a path through the obscuring material.