When the USDA published their ORAC test results in 2010, only a handful of breads were contained in the report. For that reason, we don't specifically how each different grain ranks as there are so many out there, that would require several dozen tests.
What we do know though is that breads with whole grains have moderately reasonable levels of antioxidants. We presume the reason for this is because the nutrients in grain - whether it be wheat, rye, barley, rice, sorghum, or some other - are most heavily concentrated in the outer skin. The problem with highly refined breads such as white is that they strive to reduce the amount of shell in the flour as much as possible, which leaves very little nutritional content and not to mention, a product with a high glycemic index since the fibers are largely removed.
To get the most antioxidants, vitamins, and minerals from your bread as possible, try and stick with a whole grain bread or a less-refined multi-grain variety.
USDA Database for the Oxygen Radical Absorbance Capacity (ORAC) of Selected Foods, Release 2 - Prepared by Nutrient Data Laboratory, Beltsville Human Nutrition Research Center (BHNRC), Agricultural Research Service (ARS), U.S. Department of Agriculture (USDA) - May 2010