In this paper, the filter-and-forward (FF) relay design for multiple-input-multiple-output (MIMO) orthogonal frequency-division multiplexing (OFDM) systems is investigated. Due to the considered MIMO structure, the FF relay design is investigated in the framework of joint design together with the linear MIMO transceiver. As the design criterion, first, the minimization of weighted sum mean square error (MSE) is considered. The joint design in this case is approached based on alternating optimization that iterates between the optimal design of the FF relay for given MIMO precoding and decoding matrices and the optimal design of MIMO precoding and decoding matrices for a given FF relay filter. Second, a more advanced problem of joint design for rate maximization is considered. The second problem is approached based on the obtained result regarding the first problem of weighted sum MSE minimization 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 method and significant performance improvement by the proposed FF relay over widely considered simple AF relays for MIMO-OFDM systems.