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Single crystals of Ba(Fe_1-xRu_x)₂As₂ were grown out of self flux using conventional high- temperature solution growth techniques, a T- x (0 ≤ x ≤ 0.425) phase diagram is summarized using the data derivated from temperature dependent resistances. The longitudinal in-plane magnetoresistance (LMR) has been measured on different superconducting Ba(Fe_1-xRu_x)₂As₂ single crystals, for all the samples, a negtive LMR was found in the paramagnetic state whose magnitude increases as H^₂, the suppression of spin fluctuations by the applied magnetic field could account for this minus LMR. The temperature dependent LMR coefficient curves are found to shift down systematically with the increasing doping level x, which shows the detailed procedure of suppression and the existence of drastic magnetic fluctuation in the underdoped samples. The analysis of its T dependence in an 2D itinerant nearly antiferromagnetic metal model evidences that the mode coupling coefficient a(T) exhibits a qualitative change of T variation with increasing Ru content. The latter occurs exactly at quantum critical point (QCP) where the antiferromagnetic ground state is completely suppressed, indicating the change of interaction between the conduction electrons and the fluctuating magnetic moments. LMR provides a powerful tool for studying the coupling effect between the charge carriers and the spin degrees of freedom in such iron pnictide materials.