We have searched for radio pulsations toward 49 Fermi Large Area Telescope (LAT) 1FGL Catalog γ-ray sources using the Green Bank Telescope at 350 MHz. We detected 18 millisecond pulsars (MSPs) in blind searches of the data; 10 of these were discoveries unique to our survey. 16 are binaries, with eight having short orbital periods P B < 1 day. No radio pulsations from young pulsars were detected, although three targets are coincident with apparently radio-quiet γ-ray pulsars discovered in LAT data. Here, we give an overview of the survey and present radio and γ-ray timing results for the 10 MSPs discovered. These include the only isolated MSP discovered in our survey and six short-P B binary MSPs. Of these, three have very-low-mass companions (M c ≪ 0.1 M ⊙) and hence belong to the class of black widow pulsars. Two have more massive, nondegenerate companions with extensive radio eclipses and orbitally modulated X-ray emission consistent with the redback class. Significant γ-ray pulsations have been detected from nine of the discoveries. This survey and similar efforts suggest that the majority of Galactic γ-ray sources at high Galactic latitudes are either MSPs or relatively nearby nonrecycled pulsars, with the latter having on average a much smaller radio/γ-ray beaming ratio as compared to MSPs. It also confirms that past surveys suffered from an observational bias against finding short-P B MSP systems.