It is known that the estimated energy consumption of digital-to analog converters (DACs) is around 30\% of the energy consumed by analog-to-digital converters (ADCs) keeping fixed the sampling rate and bit resolution. Assuming that similarly to ADC, DAC dissipation doubles with every extra bit of resolution, a decrease in two resolution bits, for instance from 4 to 2 bits, represents a 75$\% $ lower dissipation. The current limitations in sum-rates of 1-bit quantization have motivated researchers to consider extra bits in resolution to obtain higher levels of sum-rates. Following this, we devise coarse quantization-aware precoding using few bits for the broadcast channel of multiple-antenna systems based on the Bussgang theorem. In particular, we consider block diagonalization algorithms, which have not been considered in the literature so far. The sum-rates achieved by the proposed Coarse Quantization-Aware Block Diagonalization (CQA-BD) and its regularized version (CQA-RBD) are superior to those previously reported in the literature. Simulations illustrate the performance of the proposed CQA-BD and CGA-RBD algorithms against existing approaches.