We study the propagation of narrow electromagnetic beams associated to different K-valleys, from both the armchair and zigzag edges, in a graphene-like triangular metallic photonic structure. Due to the metallic nature of the structure, the propagation of an electromagnetic wave, with wave vector close to the quasi-triangular iso-frequency contours of the Dirac cones resulting from the band trigonal warping, can be considered as being based on series of bonding or anti-bonding states, formed by the local resonance modes in the unit cells. The transmission of a narrow incident beam depends on the local structure configurations in the beam impact zone, that determine the distribution of the excited local resonance modes as well as the coupling between the latter and the incident field, thus the beam propagation in different directions. This investigation allows us to understand the interaction between an incident field and the resonance states associated to the Dirac cones at elementary structure scales reaching the size of these states. It can find application in valley photonics for transport tuning and optimization.