This paper presents experimental results on properties of waves propagating in an unbounded and a bounded photonic crystal consisting of metallic cylinders that are arranged in a triangular lattice. First, we present transmission measurements of plane waves traversing a photonic crystal. The experiments are performed in the vicinity of a Dirac point, i.e., an isolated conical singularity of the photonic band structure. There, the transmission shows a pseudodiffusive 1/L dependence, with L being the thickness of the crystal, a phenomenon also observed in graphene. Second, eigenmode intensity distributions measured in a microwave analog of a relativistic Dirac billiard, a rectangular microwave billiard that contains a photonic crystal, are discussed. Close to the Dirac point, states have been detected that are localized at the straight edge of the photonic crystal corresponding to a zigzag edge in graphene.