In this paper, the filter-and-forward (FF) relay design for multiple-input multiple-output (MIMO) orthogonal frequency-division multiplexing (OFDM) systems is considered. Due to the considered MIMO structure, the problem of joint design of the linear MIMO transceiver at the source and the destination and the FF relay at the relay is considered. As the design criterion, the minimization of weighted sum mean-square-error (MSE) is considered first, and the joint design in this case is approached based on alternating optimization that iterates between optimal design of the FF relay for a given set of MIMO precoder and decoder and optimal design of the MIMO precoder and decoder for a given FF relay filter. Next, the joint design problem for rate maximization is considered based on the obtained result regarding weighted sum MSE and the existing result regarding the relationship between weighted MSE minimization and rate maximization. Numerical results show the effectiveness of the proposed FF relay design and significant performance improvement by FF relays over widely-considered simple AF relays for MIMO-ODFM systems.