This study considers dynamics generated by a three-species Lotka-Volterra competitive model with two discrete delays. The associated characteristic equation is a cubic exponential polynomial. Assuming the stability of the three-species positive stationary point in the no-delay model, we construct a stability switching curve on which the characteristic equation has a purely imaginary root. Thus, the stability may be lost. It is numerically confirmed that the stationary point bifurcates to a limit cycle via a supercritical Hopf bifurcation when the delay crosses the stability switching curve. It is also demonstrated that as the delay gets larger, two of three species are active, and the remaining one is inactive along the cycle. The birth of complicated dynamics will be discussed in our future research.