Using Decision Tree methodology to identify Independent Variables for Multiple Regression
Decision trees are useful for determining nested/interactive relationships between combinations of IVs and a DV. 

The model you specified, a multiple regression, presupposes a relationship between the IVs and the DV (e.g. linear). 

As you are aware, these models are different. So using a decision tree coupled with some importance measure to find predictive variables won't necessarily provide you with an optimal set of IVs in a regression model.

That being said, it can be a helpful exercise to inform you of non-linear relationships or interaction terms that could be predictive, and which may not be captured by specifying a model such as a multiple regression. 

If I were you, I wouldn't solely rely on using decision trees to determine a  set of IVs for a regression model. I would investigate penalized regression methods such as LASSO or ridge regression to help take you from a reduced candidate set of IVs to your final IVs. In addition, you might want to explore associative metrics related to your model specification that might be useful in exploring the relationships in your data, such as information values, chi-square tests, correlations, etc.

This may be helpful:
https://stats.stackexchange.com/questions/47367/decision-tree-as-variable-selection-for-logistic-regression