Free space optical (FSO) communication has been receiving increasing attention in recent years with its ability to achieve ultra-high data rates over unlicensed optical spectrum. A major performance limiting factor in FSO systems is atmospheric turbulence which severely degrades the system performance. To address this issue, multiple transmit and/or receive apertures can be employed, and the performance can be improved via diversity gain. In this paper, we investigate the bit error rate (BER) performance of FSO systems with transmit diversity or receive diversity with equal gain combining (EGC) over atmospheric turbulence channels described by the Double Generalized Gamma (Double GG) distribution. The Double GG distribution, recently proposed, generalizes many existing turbulence models in a closed-form expression and covers all turbulence conditions. Since the distribution function of a sum of Double GG random variables (RVs) appears in BER expression, we first derive a closed-form upper bound for the distribution of the sum of Double GG distributed RVs. A novel union upper bound for the average BER as well as corresponding asymptotic expression is then derived and evaluated in terms of Meijers G-functions.