True-time-delay (TTD) arrays can implement frequency-dependent rainbow beams and enable fast beam alignment in wideband millimeter-wave (mmWave) systems. In this paper, we consider the design of 3D rainbow beams for TTD-based beam training using planar arrays which are of great interest in practical deployments. We firstly design a codebook for a rectangular TTD array to realize beams that scan the entire 3D sphere using one Orthogonal Frequency Division Multiplexing (OFDM) symbol. Specifically, we determine a proper configuration of delays in TTD circuits and the minimum number of OFDM subcarriers that guarantee that all angular directions are probed at once using frequency-dependent beams. We then propose a frequency-domain power-based beam training algorithm to estimate the dominant propagation direction in terms of the azimuth and elevation angles. Lastly, we numerically evaluate the performance of the proposed algorithm in realistic mmWave channels as a function of the design and system parameters.