We report on an energy-sensitive imaging detector for studying the fragmentation of polyatomic molecules in the dissociative recombination of fast molecular ions with electrons. The system is based on a large area (10 cm x 10 cm) position-sensitive, double-sided Si-strip detector with 128 horizontal and 128 vertical strips, whose pulse height information is read out individually. The setup allows to uniquely identify fragment masses and is thus capable of measuring branching ratios between different fragmentation channels, kinetic energy releases, as well as breakup geometries, as a function of the relative ion-electron energy. The properties of the detection system, which has been installed at the TSR storage ring facility of the Max-Planck Institute for Nuclear Physics in Heidelberg, is illustrated by an investigation of the dissociative recombination of the deuterated triatomic hydrogen cation D2H+. A huge isotope effect is observed when comparing the relative branching ratio between the D2+H and the HD+D channel; the ratio 2B(D2+H)/B(HD+D), which is measured to be 1.27 +/- 0.05 at relative electron-ion energies around 0 eV, is found to increase to 3.7 +/- 0.5 at ~5 eV. Comment: 11 pages, 12 figures, submitted to Physical Review A