In this article, we propose a new approach for solving fractional differential equations based on a fractional complex transform and apply it to solve the nonlinear partial space-time Burgers equation and the space-time fractional Boussinesq equation. As a result, some new exact solutions for them are obtained. The exp-function method for partial differential equations of integer-order is directly extended to derive explicit and exact solutions of the fractional differential equations. This method can be suitable and more powerful for solving other kinds of nonlinear fractional PDEs arising in mathematical physics. The fractional derivatives are described in the modified Riemann-Liouville sense.