Constructing permutation polynomials is a hot topic in finite fields. Recently, huge kinds of permutation polynomials over ${\mathbf{F}}_{q^2}$ have been studied. In this paper, by using AGW criterion and piecewise method, we construct several classes of permutation polynomials over ${\mathbf{F}}_{q^3}$ of the forms similar to $(x^{q^2}+x^q+x+\delta {)}^{\frac{q^3-1}{d}+1}+L(x)$, for $d=2,3,4,6,$ where $L(x)$ is a linearized polynomial over ${\mathbf{F}}_q$.