How to use / interpret the coefficients from a regression model with categorical variables to get predicted variables depends on how your variables are coded. There are many different coding schemes (see here for a good overview). It sounds like you used 'reference cell coding', which most people call 'dummy coding'. I gather your race1 category is the reference category. In this case, the intercept is the mean of the race1 group. To compute the predicted value, you would solve the equation using whatever values for other variables apply and omitting the coefficients for the other categories (i.e., race2 & race3). There is some good, relevant info here, and here.
edit: The way the question is phrased made me think about situations in which there is only one factor in the model, however, @Michelle raises the question of the more general case. To keep this relatively simple, imagine a case with just two factors, e.g. race and sex, plus some continuous covariates. Using reference cell coding, we will create a dummy for male. Now, solving the regression equation without including any of the factor coefficients (i.e., just the intercept + continuous covariates) yields the predicted mean of the reference cell, which in this case is the race1 female group. Should you want to know the value for race1 males, you would solve as above, but also include the coefficient for male. If you wanted to ignore sex, or make a prediction for a mixed-sex group, you would calculate a weighted average of the above two predictions. Obviously, this will get more complicated as the number of factors, $J$, increases, but the pattern should be clear enough.