We study the phase diagram of the frustrated Heisenberg model on the triangular lattice with nearest- and next-nearest-neighbor spin-exchange coupling, on three-leg ladders. Using the density-matrix renormalization-group method, we obtain the complete phase diagram of the model, which includes quasi-long-range 120° and columnar order, and a Majumdar-Ghosh phase with short-ranged correlations. All these phases are nonchiral and planar. We also identify the nature of phase transitions.