The laser-damage resistance of multilayer-dielectric (MLD) pulse compressor gratings currently limits the energy performance of the petawatt-class OMEGA EP laser system at University of Rochester's Laboratory for Laser Energetics. The cleanliness of these components is of paramount importance; contaminants can act as absorbers during laser irradiation, initiating intense local heating and catastrophic laser-induced damage. Unfortunately, some of the most effective cleaning methods for MLD gratings--usually involving high temperatures and strong acids or bases--can themselves induce chemical degradation and thermal stresses, leading to coating delamination and defects. This work explores ways to improve the laser-damage resistance of MLD gratings through modifications to the final cleaning phase of the manufacturing process. Processes of defect formation are investigated through a combination of chemical cleaning experiments, microscopy, and modeling. We use a fracture-mechanics approach to formulate a mechanism for the initiation of micrometer-scale delamination defects that are commonly observed after chemical cleaning. The stress responses of MLD coatings to elevated-temperature chemical cleaning are estimated using a thermomechanical model, enabling us to study the effects of substrate thickness, solution temperature, and heating rates on coating stresses (and thus the risk of stress-induced failure). Finally, a low-temperature chemical cleaning approach is developed to improve laser-damage resistance while avoiding defect formation and mitigating coating stresses. We find that grating coupons cleaned using the optimized method consistently meet OMEGA EP requirements on diffraction efficiency and 1054-nm laser-damage resistance at 10 ps.