There are a variety of potential issues with this sort of decomposition. The first that comes to mind is that by "detrending" the response variables, you may be washing out some of the effect of the predictors. If X (predictor) has a trend and Y (response) depends on X, then is the trend you see in Y a response to the trend in X, is it an intrinsic trend in Y, or a mix? (I suspect this is more of a problem if you are trying to prove significance of X as a predictor of Y, or estimate regression coefficients, than if you are just trying to forecast Y.)
For a single predictor (simple regression), I'd be tempted to use an ARIMA transfer function model. I'm not sure if transfer functions generalize to multiple predictors (it's been years since I looked at time series methods).