Massive MIMO systems achieve high sum spectral efficiency by simultaneously serving large numbers of users. In time division duplexing systems, however, the reuse of uplink training pilots among cells results in channel estimation errors, which causes downlink inter-cell interference. Handling this interference is challenging due to the large channel dimensionality and the high complexity associated with implementing large precoding/combining matrices. In this paper, we propose multi-layer precoding to enable efficient and low-complexity operation in full-dimensional massive MIMO, where a large number of antennas are used in two dimensions. In multi-layer precoding, the precoding matrix of each base station is written as a product of a number of precoding matrices. Multi-layer precoding: 1) leverages the directional characteristics of large-scale MIMO channels to manage inter-cell interference with low channel knowledge requirements and 2) allows for an efficient implementation using hybrid analog/digital architectures. We present and analyze a specific multi-layer precoding design for full-dimensional massive MIMO systems. The asymptotic optimality of the proposed design is then proved for some special yet important channels. Numerical simulations verify the analytical results and illustrate the potential gains of multi-layer precoding compared with other multi-cell precoding solutions.