If you google: working out while on period; you will find a whole world of women out there with the same question! There are a lot of misconceptions about working out while menstruating, and many of them are just myths perpetuated from years back when women were considered unclean and shunned during that time of the month.
There is absolutely no evidence to support that you shouldn't be able to work out during your cycle. In fact, many studies show that light exercise, cardio, or strength training releases endorphins, which actually help relieve menstrual cramping and pain. A little sweat session can also help with bloating and releasing retained fluid.
If you have very heavy or painful periods, you may want to take a day or two off. The best suggestion from gathered evidence, would be to take one day off, if needed, and then try easy to moderate exercise until your symptoms get better. Drink plenty of water and keep a clean and healthy diet.