Cook rice according to package instructions. Meanwhile, in a blender, blend herbs, oil, and vinegar until roughly chopped, scraping down sides as needed. With machine running, gradually add 1/4 to 1/3 cup cold water and blend until dressing is smooth and pourable. Transfer to a small bowl and season with salt and pepper.
In a large bowl, toss arugula with three-quarters the dressing. Transfer to a serving platter. In bowl, stir together rice, chicken, scallions, tomatoes, and remaining dressing. Season to taste with salt and pepper and arrange on top of arugula.