The most relevant linear precoding method for frequency-flat MIMO broadcast channels is block diagonalization (BD) which, under certain conditions, attains the same nonlinear dirty paper coding channel capacity. However, BD is not easily translated to frequency-selective channels, since space-time information is required for transceiver design. In this paper, we demonstrate that BD is feasible in frequency-selective MIMO broadcast channels to eliminate inter-user interference (IUI) if the transmit block length is sufficiently large, and if the number of transmit antennas is greater than the number of users. We also propose three different approaches to mitigate/eliminate inter-symbol interference (ISI) in block transmissions: i) time-reversal-based BD (TRBD) which maximizes spatial focusing around the receivers using transmitter processing only, ii) equalized BD (EBD) which minimizes the ISI using transmitter processing only, and iii) joint processing BD (JPBD), which uses linear processing at the transmitter and the receiver to suppress ISI. We analyze the theoretical diversity and multiplexing gains of these techniques, and we demonstrate that JPBD approximates full multiplexing gain for a sufficiently large transmit block length. Extensive numerical simulations show that the achievable rate and probability of error performance of all the proposed methods improve those of conventional time-reversal beamforming. Moreover, JPBD provides the highest achievable rate region for frequency-selective MIMO broadcast channels.