We propose an analytical approach to evaluate the performance of space-diversity free-space optical (FSO) systems over correlated Gamma-Gamma (ΓΓ) fading channels. To do this, we firstly derive an infinite series representation for the moment generating function (MGF) of the sum of arbitrarily correlated ΓΓ random variables (RVs). Then, a closed-form approximate expression for the probability density function (PDF) of the sum of ΓΓ RVs is obtained from the MGF with the aid of Padé approximation. The performance of the FSO system is then evaluated based on this PDF. Through some numerical results we illustrate the accuracy of the proposed method and further discuss its limitations.