In this paper, we consider precoder design for multiuser multiple-input-multiple-output (MIMO) Gaussian broadcast (BC) channels and propose a generalized zero-forcing (GZF) precoder based on successive dirty-paper coding (DPC), i.e., the GZF-DP precoder. The GZF-DP precoder is an extension of the GZF-DP precoder designed earlier for multi-input-single-output broadcast (MISO-BC) channels, and also a generalization of both the linear block-diagonalization ZF (BD-ZF) and the successive ZF with DPC (SZF-DPC) precoders. With the GZF-DP precoder, the depth of the inter-user interference after precoding (and before the DPC) can be specified at will by a parameter ν, which provides a trade-off between the optimal rates and the DPC implementation-complexity. Utilizing DPC, the known non-causal inter-user interferences from the other (up to) ν users are canceled through successive encoding. Within the class of GZF-DP, we analyze the optimal precoder designs both for sum-rate and minimum user-rate maximizations, which are solved in closed-forms in conjunction with water-filling algorithms depending on ν. We show through numerical results that, the proposed GZF-DP precoder with a small ν renders significant rate increments compared to the linear BD-ZF precoder, and is close to the SZF-DP preocder with a much less DPC complexity.