Block diagonalization (BD) is an attractive method for approaching sum capacity in multiuser multiple-input multiple-output (MIMO) broadcast channels. However, when channel state information (CSI) is limited, the performance of the BD is degraded because inter-user interference cannot be completely eliminated. In this paper, we study limited feedback systems for the BD where a large number of users are supported per cell. Based on an estimate of the received signalto-interference-plus-noise ratio (SINR), we present an efficient feedback scheme which reflects the quality of each user's channel in addition to the use of a quantization codebook. In simulation, we compare the sum rate of the BD scheme under practical scenarios and verify that the proposed SINR feedback is an effective way to exploit multiuser diversity.