Massive multi-user multiple-input multiple-output (MU-MIMO) systems enable high spatial resolution, high spectral efficiency, and improved link reliability compared to traditional MIMO systems due to the large number of antenna elements deployed at the base station (BS). Nevertheless, conventional massive MU-MIMO BS transceiver designs rely on centralized linear precoding algorithms, which entail high interconnect data rates and a prohibitive complexity at the centralized baseband processing unit. In this paper, we consider an MU-MIMO system, where each user device is served with multiple independent data streams in the downlink. To address the aforementioned challenges, we propose a novel decentralized BS architecture, and develop a novel decentralized precoding algorithm based on eigen-zero-forcing (EZF). Our proposed approach relies on parallelizing the baseband processing tasks across multiple antenna clusters at the BS, while minimizing the interconnection requirements between the clusters, and is shown to closely approach the performance of centralized EZF.