The joint design of transmitter and receiver for multichannel data transmission over dispersive channels is considered. In particular, the practically important case where the transmitter consists of FIR filters and the channel impulse response has arbitrary length is addressed. The design criterion is the minimization of the mean squared error at the receiver output under the constraint of a fixed transmit power. The proposed algorithm allows a straightforward transmitter design and yields (in general) a near-optimal solution for the transmit filters. Under certain conditions, the exact solution for the optimal transmitter is obtained.