In this letter, we consider linear precoding for downlink massive multi-user (MU) multiple-input multiple-output (MIMO) systems. We propose the novel successively-regularized zero forcing (SRZF) precoding, which exploits successive null spaces of the MIMO channels of the users, along with regularization, to control the inter-user interference (IUI) and to enhance performance and robustness to imperfect channel state information (CSI) at the base station (BS). We study the IUI characteristics of the proposed SRZF precoding for perfect and imperfect CSI at the BS. Furthermore, via computer simulations, we compare the sum rate of SRZF precoding with those of several baseline schemes including conventional and regularized zero forcing (ZF) precoding. Our simulation results reveal that, for massive MIMO systems with inter-user channel correlations, the proposed SRZF precoding significantly outperforms the considered baseline schemes for both perfect and imperfect CSI at the BS.