We have used two methods to search for surviving companions of Type Ia supernova progenitors in three Balmer-dominated supernova remnants in the Large Magellanic Cloud: 0519-69.0, 0505-67.9 (DEM L71), and 0548-70.4. In the first method, we use the Hubble Space Telescope photometric measurements of stars to construct color-magnitude diagrams (CMDs) and compare positions of stars in the CMDs with those expected from theoretical post-impact evolution of surviving main-sequence or helium star companions. No obvious candidates of surviving companions are identified in this photometric search. Future models for surviving red giant companions or with different explosion mechanisms are needed for thorough comparisons with these observations in order to make more definitive conclusions. In the second method, we use Multi Unit Spectroscopic Explorer observations of 0519-69.0 and DEM L71 to carry out spectroscopic analyses of stars in order to use large peculiar radial velocities as diagnostics of surviving companions. We find a star in 0519-69.0 and a star in DEM L71 moving at radial velocities of 182+/-0km/s and 213+/-0km/s, respectively, more than 2.5{sigma} from the mean radial velocity of the underlying stellar population, 264 and 270km/s, respectively. These stars need higher-quality spectra to investigate their abundances and rotation velocities to determine whether they are indeed surviving companions of the supernova progenitors.