A Mode-III crack problem in a functionally graded material modeled by anisotropic strain-gradient elasticity theory is solved by the integral equation method. The gradient elasticity theory has two material characteristic lengths $\u2113$ and $\u2113\u2032$, which are responsible for volumetric and surface strain-gradient terms, respectively. The governing differential equation of the problem is derived assuming that the shear modulus $G$ is a function of $x$, i.e., $G=G(x)=G0e\beta x$, where $G0$ and $\beta $ are material constants. A hypersingular integrodifferential equation is derived and discretized by means of the collocation method and a Chebyshev polynomial expansion. Numerical results are given in terms of the crack opening displacements, strains, and stresses with various combinations of the parameters $\u2113$, $\u2113\u2032$, and $\beta $. Formulas for the stress intensity factors, $KIII$, are derived and numerical results are provided.