  We consider a vibrating triangular mass-truss lattice whose unit cell
contains a resonator of a triangular shape. The resonators are connected to the
triangular lattice by trusses. Each resonator is tilted, i.e. it is rotated
with respect to the triangular lattice's unit cell through an angle
$\vartheta_0$. This geometrical parameter is responsible for the emergence of a
resonant mode in the Bloch spectrum for elastic waves and strongly affects the
dispersive properties of the lattice. Additionally, the tilting angle
$\vartheta_0$ triggers the opening of a band gap at a Dirac-like point. We
provide a physical interpretation of these phenomena and discuss the dynamical
implications on elastic Bloch waves. The dispersion properties are used to
design a structured interface containing tilted resonators which exhibit
negative refraction and focussing, as in a "flat elastic lens".
