The implementation of a cloud radio access network (C-RAN) with full dimensional (FD) multiple-input multiple-output (MIMO) is faced with the challenge of controlling the fronthaul overhead for the transmission of baseband signals as the number of horizontal and vertical antennas grows larger. This paper proposes to leverage the special low-rank structure of the FD-MIMO channel, which is characterized by a time-invariant elevation component and a time-varying azimuth component, by means of a layered precoding approach, to reduce the fronthaul overhead. According to this scheme, separate precoding matrices are applied for the azimuth and elevation channel components, with different rates of adaptation to the channel variations and correspondingly different impacts on the fronthaul capacity. Moreover, we consider two different central unit (CU)-radio unit (RU) functional splits at the physical layer, namely, the conventional C-RAN implementation and an alternative one in which coding and precoding are performed at the RUs. Via numerical results, it is shown that the layered schemes significantly outperform conventional nonlayered schemes, particularly in the regime of low fronthaul capacity and a large number of vertical antennas.