Recently, a number of techniques have been introduced to exploit multiuser diversity of a wireless multiple-input multiple-output (MIMO) broadcast channel (BC) that consists of a base station with t transmit antennas and K users with multiple antennas. However, prior works have ignored the rate overhead associated with feedback of MIMO BC channel state information at transmitter (CSIT), which is roughly K times larger than single-user MIMO CSIT (i.e., it is O(tr) where r = rk and rk is the number of antennas at the kth user). Considering the amount of feedback signaling, quantization is a necessity for effective feedback transmission as a form of partial CSIT. In this paper, we propose the greedy multi-channel selection diversity (greedy MCSD) scheme based on block MMSE QR decomposition with dirty paper coding (block MMSE-DP), where partial CSIT is almost sufficient. The sum-rate performance of our novel scheme approaches extremely close to the sum capacity of MIMO BC as the number of users increases, whereas the feedback overhead is reduced by a factor of 2t3/L(t2-t), in which L is the number of active channel vectors. Simulation results validate the expectation from the analysis. In addition, the proposed scheme is shown to be appropriate for reconfigurable implementation.