Most Galactic point sources of gamma rays remain unidentified. The few (extrasolar) sources that have been identified are all young, rotation-powered pulsars, all but one of which were identified using radio ephemerides. The radio-quiet Geminga pulsar was identified only after pulsations were discovered in a coincident X-ray source. Observational evidence indicates that many of the unidentified Galactic sources are likely to be pulsars, and some theoretical models predict a potentially large population of radio-quiet gamma-ray pulsars. We present a new method for performing sensitive gamma-ray pulsar searches. We used this method to search several of the strongest EGRET sources for pulsations. This was a blind search for new pulsars, covering a frequency and a frequency-derivative phase space large enough to detect Crab-like pulsars as well as lower frequency, high magnetic field magnetars. No new pulsars were discovered, and we report upper limits constraining the characteristics of any signals contained in the data sets searched.