I have ovarian cysts on both ovaries since a hysterectomy...will the doctor remove them?

Depends: At 39 years old, you probably have functioning ovaries. If they are removed you are (without hormonal replacement) plunged into menopause. A question would be, are you in pain, or are the ovaries just known to have asyptomatic cysts on them? Most physicians would remove symptomatic ovarian cysts if other means to shrink them didm't work.

Possibly: Since you're in your reproductive years, you may form a cyst persisting from a recent ovulation that will then resolve on its own. An ultrasound should be performed in 6-8 weeks. If the cyst does not resolve then surgery, usually minimally invasive laparoscopic is necessary to remove the cyst and/or involved ovary. The tissue will be sent to pathology to determine if cancerous (unlikely) or not.