I'm a big fan of euphemisms, but who gets what depends on who I am talking to. "That time", "It's that time" is the most direct way I will say it unless I'm talking to a doctor. A doctor would get "my period". When talking to family or other people I don't like discussing health matters with in much detail, I might just say I'm not feeling well and if they ask for more, vaguely indicate. My partner and I use slightly more colorful things, but I'll keep them between us. :3
I don't think it is widely used, but I have heard of "Flag Day" being something the Japanese sometimes say. If you don't know the Japanese flag well, go look at it and figure it out!