The acyclic chromatic index of a graph $G$ is the least number of colors needed to properly color its edges so that none of its cycles is bichromatic. In this work, we show that $2Δ-1$ colors are sufficient to produce such a coloring, where $Δ$ is the maximum degree of the graph. In contrast with most extant randomized algorithmic approaches to the chromatic index, where the algorithms presuppose enough colors to guarantee properness deterministically and use randomness only to deal with the bichromatic cycles, our randomized, Moser-type algorithm produces a not necessarily proper random coloring, in a structured way, trying to avoid cycles whose edges of the same parity are homochromatic, and only when this goal is reached it checks for properness. It repeats until properness is attained.