  We propose exact quantization conditions for the quantum integrable systems
of Goncharov and Kenyon, based on the enumerative geometry of the corresponding
toric Calabi-Yau manifolds. Our conjecture builds upon recent results on the
quantization of mirror curves, and generalizes a previous proposal for the
quantization of the relativistic Toda lattice. We present explicit tests of our
conjecture for the integrable systems associated to the resolved C^3/Z_5 and
C^3/Z_6 orbifolds.
