An effective mechanism for dissipative soliton acceleration in nonlinear optical lattices under the presence of linear gain and nonlinear loss is presented. The key idea for soliton acceleration consists of the dynamical reduction of the amplitude of the effective potential experienced by the soliton so that its kinetic energy eventually increases. This is possible through the dependence of the effective potential amplitude on the soliton mass, which can be varied due to the presence of gain and loss mechanisms. In contrast to the case where either the linear or the nonlinear refractive index is spatially modulated, we show that when both indices are modulated with the same period we can have soliton acceleration and mass increasing as well as stable soliton propagation with constant non-oscillating velocity. The acceleration mechanism is shown to be very robust for a wide range of configurations.