There are plenty of examples (shift workers for examples) of people living with time shifting days or days that aren't 24 hours long. There are some signs of stress and depression from that but nothing long term damaging. Within a few generations you can expect most people to have adapted.
Most likely people would adjust to a 27 hour day, with three local days being compressed into one "day". That adjustment wouldn't take very long at all.
Women's periods would be unlikely to change (although with the science level involved most likely women would no longer have periods except when wanting a child), and gestation times would almost certainly not change. They are both tied far more to biological facts like how long it takes an embryo to grow than to anything else.