Random unitary beamforming is one of the schemes reducing the amount of feedback information in multiuser diversity techniques with multiple-antenna downlink transmission. In Multiple-Input Multiple-Output (MIMO) systems, throughput performance is greatly improved using AMC (Adaptive Modulation and Coding). Throughput performance is also improved by allocating power among streams appropriately. In random unitary beamforming, transmitter has only partial channel state information (CSI) of each receiver. Thus, it is difficult for random unitary beamforming to use conventional power allocation methods that use full CSI at all the receivers. In this paper, we propose a new scheduling algorithm with power allocation for downlink random unitary beamforming that improves throughput performance without full CSI. We provide numerical results of the proposed scheduling algorithm and compare them to those of the conventional random unitary beamforming scheduling algorithm.