In a large skillet over medium-high heat, heat oil. Add chicken; season with oregano, salt, and pepper. Cook, turning halfway through, until golden brown and an instant-read thermometer inserted into thickest part registers 165°, about 8 minutes per side. Transfer to a cutting board. Let rest 5 minutes, then thinly slice.
Divide quinoa among bowls. Top with cucumber, avocado, tomatoes, olives, feta, dill, and chicken. Drizzle with dressing.
