In a food processor, puree the cilantro stems, garlic, oyster sauce, half of the lime juice and 1/2 tsp. each salt and pepper. Transfer to a bowl, add the chicken and toss to coat. Cover and refrigerate for 3 hours or overnight.
Position a rack in the upper third of the oven and preheat to 400 degrees . Place the chicken on a parchment-lined baking sheet and bake, turning once, until cooked through, 45 to 50 minutes.
Meanwhile, in a medium saucepan, combine the rice, 2 1/4 cups water and 1/2 tsp. salt over medium-low heat. Cover and cook until the rice is tender, about 20 minutes. Let sit, covered, for 10 minutes; fluff with a fork.
In a large bowl, combine the cucumbers, cilantro leaves and remaining lime juice; season with salt and pepper. Serve with the chicken and rice.
