Gicumbi is a district in Northern Province, Rwanda. Its capital is Byumba, which is also the provincial capital. The district lies due-north of Kigaliā¦Gicumbi is a district in Northern Province, Rwanda. Its capital is Byumba, which is also the provincial capital. The district lies due-north of Kigali, straddling the major road from Kigali to Kampala. It is a hilly district.