In this study, the nonlinear fractional partial differential equations have been defined by the modified Riemann-Liouville fractional derivative. By using this fractional derivative and traveling wave transformation, the nonlinear fractional partial differential equations have been converted into nonlinear ordinary differential equations. The modified trial equation method is implemented to obtain exact solutions of the nonlinear fractional Klein-Gordon equation and fractional clannish random walker's parabolic equation. As a result, some exact solutions including single kink solution and periodic and rational function solutions of these equations have been successfully obtained. Copyright (c) 2015 John Wiley & Sons, Ltd.