Precoding with block diagonalization is an attractive approach for approaching sum capacity in multiuser MIMO (multiple input multiple output) broadcast channels. This method though requires either global channel state information at every receiver or an additional training phase, which requires additional system planning. This paper proposes a lattice based multi-user precoder that uses block diagonalization combined with pre-equalization and perturbation for the multiuser MIMO broadcast channel. Achievable rates are computed and used to show that the proposed technique approaches the capacity with block diagonalization and water-filling but does not require the additional channel state information at the receiver. Monte Carlo simulations under the case of equal power allocation show that the proposed method provides better diversity and BER (bit error rate) performance than block diagonalization with a zero-forcing receiver.