We present measurements of three-particle correlations for various harmonics in Au+Au collisions at energies ranging from root s(NN) = 7.7 to 200 GeV using the STAR detector. The quantity < cos(m phi(1) + n phi(2) - (m + n)phi(3))>, with phi being the azimuthal angles of the particles is evaluated as a function of root s(NN), collision centrality, transverse momentum, P-T, pseudorapidity difference, Delta(eta), and harmonics (m and n). These data provide detailed information on global event properties such as the three-dimensional structure of the initial overlap region, the expansion dynamics of the matter produced in the collisions, and the transport properties of the medium. A strong dependence on Delta(eta) is observed for most harmonic combinations, which is consistent with breaking of longitudinal boost invariance. An interesting energy dependence is observed when one of the harmonics m, n, or m n is equal to two, for which the correlators are dominated by the two-particle correlations relative to the second-harmonic event plane. These measurements can be used to constrain models of heavy-ion collisions over a wide range of temperature and baryon chemical potential.