The author considers an extension of the concept of least absolute deviation regression for problems with multivariate response. The approach is based on a transformation and retransformation technique that chooses a data-driven coordinate system for transforming the response vectors and then retransform 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), is suggested, which adaptively chooses the optimal data-driven coordinate system and then computes the regression estimates. Resampling techniques like the bootstrap are used to conveniently estimate the standard errors of TREMMER estimates. 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 adaptive transformation-retransformation median estimate are also discussed. The performance of the procedure is demonstrated with two real examples.