A vertex coloring of a graph G is called acyclic if no two adjacent vertices have the same color and no cycle in G is bichromatic. The acyclic chromatic number a(G) of a graph G is the least number of colors in an acyclic coloring of G. In this paper, we obtain bounds for the acyclic chromatic number of the strong product of trees and graphs. An exact value for the acyclic chromatic number of the strong product of two paths and the acyclic chromatic number of strong product of a path and a tree are also given. Further observations are made on the upper bound for the strong product of three paths.