Successive zero-forcing (SZF) is a precoding technique for broadcast channels that partially precancels the multiuser interference. Due to the residual interference, the optimal precoder design for SZF is difficult to find, and currently unknown. This letter proposes a suboptimal iterative precoder design for the sum rate maximization problem for SZF, using a local optimization method. In each iteration, the precoders are updated to maximize a first-order approximation of the sum rate, which can be easily solved since the approximation is concave with the precoders. By initializing the precoders with zero matrices, we analytically prove that the iterative design can achieve a larger sum rate than the block diagonalization (BD) approach. Moreover, numerical results demonstrate the superior sum-rate performance of the proposed design over the existing scheme, which is based on a heuristic method.