We demonstrate a novel approach to achieve wavelength-tunable ultrashort pulses from an all-fiber mode-locked laser with a saturable absorber based on the nonlinear Kerr beam clean-up effect. This saturable absorber was formed by a single-mode fiber spliced to a graded-index multimode fiber, and its tunable band-pass filter effect is described by a numerical model. By adjusting the bending condition of the graded-index multimode fiber, the laser could produce dissipative soliton pulses with their central wavelength tunable from 1040 nm to 1063 nm. The pulse duration of the output laser could be compressed externally to 791 fs, and the signal to noise ratio of its radio frequency spectrum was measured to be 75.5 dB.