Without sounding like I'm completely copping out of answering this question...it's totally up to you. Everyone is so different anyway, so why does that change when we are on our period?
Some people have a fairly stress free period with little to no cramps, light bleeding that maybe only lasts a couple of days and are pretty happy to just carry on as normal. If that's you, then awesome, go for it I say.
Sadly, that's not me.
The other end of the spectrum, like me, suffer quite badly with this time of the month. Excruciating cramps, achey all over, totally fatigued, super heavy flow for what seems like weeks and just generally feeling rubbish! You of course then have everything else in between these two examples. Don't you just love being a woman!
So personally for me, the week before my period is due, I suffer with severe PMS (like crawling on the floor-bawling my eyes out-in pure agony, kind of cramps) and that's just before the actual period...yay. So during this week I find it very difficult to bring myself to exercise properly as my body is physically and psycologically exhausted. However, during this week I try to keep up whatever I can and usually end up doing some light cardio/toning or just walking and stretching. If I happen to feel good, then I make the most of it and move my body as often light exercise actually helps alleviate these horrendous cramps.
During my period is another story...
I am unlucky enough to have a very heavy period which leaves me feeling incredibly tired for just over a week (I used to get anaemia so I am prone to being slightly anaemic whilst I'm on my period, which can make me very tried). This often means the last thing I want to do is exercise. However, again I just do what I can and listen to my body. If I feel up to working out I do but if I don't, then I don't. I respect my body and and give it rest when it needs rest. That being said I don't use it as an excuse if I'm actually feeling fine. Each cycle varies. But I definitely tend to keep my workouts lower intensity than normal, if I feel ok enough to train during my period - think walking, pilates, gentle aerobics and less volume when it comes to weight training.
Like I said it is totally individual to each woman, as we are all so different and therefore I can't just give a blanket answer that will suit everyone. Sorry!
Moral of the story, listen to your body, be kind to yourself and just do what you can!