Generally, no. Some scientists believe the great apes, including chimpanzees and orangutans shed the uterine lining regulary, rather than experiencing oestreus. All female mammals come into "heat", but this varies from species to species. Many larger animals only come in to oestrus once per year and during that time, there may be some minor vaginal discharge but it's not a true"period".
Here in the western U.S., we breed cattle in the fall. This is the only time the female will accept the bull so as to have spring born calves...