We have studied the phase behavior of a two-dimensional system of Janus-like particles on a triangular lattice using the Monte Carlo method in a grand canonical ensemble. Assuming that each particle can take on only one of the six orientations, two versions of the model have been considered. In the first version, the strength of attractive interactions has been assumed to depend on the degree to which the attractive patches of neighboring particles overlap. In the second version, it has been assumed that it is the same for any mutual orientations, in which the attractive patches overlap. It has been demonstrated that both models lead to qualitatively different phase behaviors. In the case of the first model, the self-assembly leads to different stripped structures depending on the density and temperature. In particular, we have found that, at sufficiently low temperatures, condensation leads from a very dilute lamellar gas phase to a high density zigzag phase. At intermediate temperatures, the system undergoes two first-order phase transitions, while, at sufficiently high temperatures, only one continuous transition takes place. The phase diagram has been estimated. In the case of the second model, we have found only one first-order transition at low temperatures. This transition occurs between a dilute gas-like phase and the ordered phase, which forms a kagome lattice of density equal to 6/7. A further increase of the density has been demonstrated to lead to the reorientation of particles and the formation of a dense glass-like structure.