This study focuses on resource block allocation issue in the downlink transmission systems of the Long Term Evolution (LTE). In existing LTE standards, all Allocation Units (AUs) allocated to any user must adopt the same Modulation and Coding Scheme (MCS), which is determined by the AU with the worst channel condition. Despite its simplicity, this strategy incurs significant performance degradation since the achievable system throughput is limited by the AUs having the worst channel quality. To address this issue, a two-step resource block allocation algorithm is proposed in this paper. The algorithm first allocates AUs to each user according to the users' priorities and the number of their required AUs. Then, a re-allocation mechanism is introduced. Specifically, for any given user, the AUs with the worst channel condition are removed. In this manner, the users may adopt a higher MCS level, and the achievable data rate can be increased. Finally, all the unallocated AUs are assigned among users without changing the chosen MCSs, and the total throughput of the system is further enhanced. Simulation results show that thanks to the proposed algorithm, the system gains higher throughput without adding too many complexities.