I'm 45 and definitely in the midst of "peri", but for the most part...I try and not let it affect any of my workouts, be it cycling or the gym. The 2nd day is my heaviest and most troublesome, when it comes to planning bathroom stops. I generally can't go more than 2-3 hours without a tampon change on day 2. Aside from feeling a lack of energy (which in my case, is nearly everyday anyways) and perhaps a headache, neither of those things would stop me from riding on my period.
2012 Seven Axiom SL - Specialized Ruby SL 155