An extension of the concept of least absolute deviation regression for problems with multivariate response is considered. The approach is based on a transformation and retransformation technique that chooses a data-driven coordinate system for transforming the response vectors and then retransforms the estimate of the matrix of regression parameters, which is obtained by performing coordinatewise least absolute deviations regression on the transformed response vectors. It is shown that the estimates are equivariant under non-singular linear transformations of the response vectors. An algorithm called TREMMER (Transformation Retransformation Estimates in Multivariate MEdian Regression) has been suggested, which adaptively chooses the optimal data-driven coordinate system and then computes the regression estimates. We have also indicated how resampling techniques like the bootstrap can be used to conveniently estimate the standard errors of TREMMER estimates. It is shown that the proposed estimate is more efficient than the non-equivariant coordinatewise least absolute deviations estimate, and it outperforms ordinary least-squares estimates in the case of heavy-tailed non-normal multivariate error distributions. Asymptotic normality and some other optimality properties of the estimate are also discussed. Some interesting examples are presented to motivate the need for affine equivariant estimation in multivariate median regression and to demonstrate the performance of the proposed methodology.