Rinse the rice and drain well. Combine the rice and broth in a saucepan and bring to a boil. Cover and reduce heat to a low simmer. Cook until the liquid is absorbed, 40 to 45 min. Remove from heat and let stand for 5 min. Fluff with a fork and transfer to a large bowl to cool to room temperature.
In a mixing bowl, whisk together the olive oil, lemon juice and garlic. Stir in the diced cucumber, tomatoes, cooled rice, parsley, chives and mint until blended. Season with salt and pepper. Cover and refrigerate until ready to serve.