If you are referring to estrous cycles, then no, all placental mammals (neither marsupials nor monotremes) have them. However, I assume you mean overt menstrual cycles, where the endometrium gets shed when no fertilised ovum has been implanted. Most mammals simply resorb the endometrium if there has been no successful implantation.

The only mammals that I can think of offhand that have overt menstruation are the great apes, including humans. In the other great apes menstrual flow generally is much smaller, and the entire process is less dramatic than in humans, but it does happen.

Was that what you wanted to know?

I know that cats don't, they only ovulate during copulation, so if there is no sex there is no preparation for fetus and there is nothing to cast, and if there is sex there are also little kittens few months later... But my friend had a dog, well a bitch actualy of corse ;) and it (or is it she?) had to wear special panties or diapers to avoid spotting his furniture with menstrual blood.