  This article presents experimental results on properties of waves propagating
in an unbounded and a bounded photonic crystal consisting of metallic cylinders
which 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 which are localized at the straight
edge of the photonic crystal corresponding to a zigzag edge in graphene.
