I've been known to use dildos vaginally while I'm on my period, but lately I've been indulging in more anal play during my periods. It's great! I can leave my diva cup in and be completely naked with no worries about leaking.
If I did want to use a dildo, I'd just put down a dark towel, t-shirt, or blanket and go for it. Call me crazy, but I'm pretty horny while I'm on my period and I think menstrual blood is actually pretty beautiful.