This paper addresses the suboptimal energy efficiency of conventional digital precoding schemes in multiple-input multiple-output (MIMO) systems. Through an analysis of the power amplifier (PA) output power distribution associated with conventional precoders, it is observed that these power distributions can be quite uneven, resulting in large PA backoff (thus low efficiency) and high power consumption. To tackle this issue, we propose a novel approach called flat precoding, which aims to control the flatness of the power distribution within a desired interval. In addition to reducing PA power consumption, flat precoding offers the advantage of requiring smaller saturation levels for PAs, which reduces the size of PAs and lowers the cost. To incorporate the concept of flat power distribution into precoding design, we introduce a new lower-bound per-antenna power constraint alongside the conventional sum power constraint and the upper-bound per-antenna power constraint. By adjusting the lower-bound and upper-bound values, we can effectively control the level of flatness in the power distribution. We then seek to find a flat precoder that satisfies these three sets of constraints while maximizing the weighted sum rate (WSR). In particular, we develop efficient algorithms to design weighted minimum mean squared error (WMMSE) and zero-forcing (ZF)-type precoders with controllable flatness features that maximize WSR. Numerical results demonstrate that complete flat precoding approaches, where the power distribution is a straight line, achieve the best trade-off between spectral efficiency and energy efficiency for existing PA technologies. We also show that the proposed ZF and WMMSE precoding methods can approach the performance of their conventional counterparts with only the sum power constraint, while significantly reducing PA size and power consumption.