When you have multiple dependent variables, there is always a question of how you should analyze the situation. The basic question is how you understand these variables to be related to each other. If you believe they are unrelated, then you can safely run several multiple regression analyses using each DV in turn. If they aren't unrelated, then the relationships amongst them need to be taken into account somehow. If they are multiple measures of the same underlying construct, then they can be combined into a single variable and used as a single DV. For example, consider a personality questionnaire that attempts to measure extroversion. There will be a number of questions that are all understood as measuring the same thing, viz., extroversion. In this case, the responses to the various items are combined (typically by adding them up or averaging them), and you can then run a single multiple regression model with the combined score as a single DV.
The question is, whether your measures are like that. How should you determine this? First, your theory, or the main theories operative in your field, may provide an answer. If they don't, or you want to check them, you can run some analyses on your data. A factor analysis is one such possibility; if your measures load onto a single factor, you're good to go. If they load onto more than one factor, you should perform a non-orthogonal rotation such as oblimin, and check the correlations amongst the factors; if they are low enough for your satisfaction, then you can treat them as unrelated as discussed above. Another possibility is to run multiple regressions with each DV in turn and save the residuals. Then you can see if the residuals are correlated with each other; if they're not, you're fine.
If your measures are related to each other, you need to account for that somehow in your modeling. Presumably the best approach would be a structural equation model that allows you to specify the manner in which the measures are related.