We present the results of a second survey to search for maser emission from the satellite line of the hydroxyl radical (OH) at 1720.5 MHz in the direction of 75 supernova remnants (SNRs) in the Galaxy with the Parkes 64 m telescope. We detect OH (1720 MHz) emission toward 33 SNRs. Additional observations of five SNRs not previously surveyed at Parkes were made with the Australia Telescope Compact Array (ATCA), bringing the total number of SNRs surveyed to 80. Confirmation observations with the Very Large Array (VLA) and the ATCA have resulted in the detection of compact emission from four objects, three of which are SNRs, and one which is believed to be associated with an R ii region (located 12' from a remnant). We argue that the OH (1720 MHz) features are indeed masers and that they are physically associated with the remnants. The present survey brings the number of SNRs searched in our Galaxy for OH (1720 MHz) emission to 160. Of these there are a total of 17 with maser detections (10% of the sample studied). Considering the distribution of the detections as a whole, we show that they closely trace the molecular gas in the Galaxy. The available evidence supports a model in which the masers are collisionally excited by the passage of the supernova remnant shock through an adjacent molecular cloud. As such, the OH (1720 MHz) line provides a powerful new diagnostic tool for investigating the interaction of SNRs with molecular clouds, including, for example, the identification of promising candidates for high energy searches for sites of cosmic ray acceleration. We also note that many of our OH (1720 MHz) detections are drawn from the morphological class of supernova remnants producing center-brightened, thermal x rays. We suggest that the presence of dense molecular gas is required to form this distinctive morphology.