A 3-parameter generalization of the Lunin-Maldacena background has recently been constructed by Frolov. This γ_i-deformed background is non-supersymmetric. We consider strings in this γ_i-deformed × S⁵ background rotating in three orthogonal planes (the 3-spin sector) in a fast motion limit, in which the total angular momentum J is assumed to be large. We show that there exists a consistent transformation which takes the undeformed equations of motion into the γ_i-deformed equations of motion. This transformation is used to construct a Lax pair for the bosonic part of the γ_i-deformed theory in the fast motion limit. This implies the integrability of the bosonic part of the γ_i-deformed string sigma model in the fast motion limit.