The U.S. Census Bureau has started releasing data from the 2010 Census. This map shows the change in the racial/ethnic minority (i.e., anything other than non-Hispanic White) population over the last decade:
They released a report, An Overview: Race and Hispanic Origin in the 2010 Census (available here), which includes data on those who reported more than one race. Among those who reported more than one race, the vast majority listed two. Here are the most commonly reported combinations:
AIAN = American Indian/Alaska Native, NHPI = Native Hawaiian/Pacific Islander, and SOR = some other race.
Laura E. pointed out that New Geography posted some maps based on 2010 Census data. Here’s the Hispanic population as a percent of the total population, by county (notice that the legend need to be multiplied by 100 to get the percent):
The African American population (alone or in combination with another race, and again, multiply by 100):