Studies of Galactic supernova remnants (SNRs) are typically impeded by such factors as extensive absorption along Galactic lines of sight and significant uncertainties in the distances to these sources. To help address these known shortcomings and to explore several issues related to SNR evolution (such as SNR-interstellar medium interactions and cosmic-ray acceleration), we have analyzed original and archived Chandra observations of a sample of nearby face-on spiral galaxies to search for X-ray counterparts to known SNRs. Results of this search will be presented, including discussions of wavelength-dependent selection effects in searches for SNRs and the possible detection of several X-ray binary/SNR systems in these galaxies.