From what I have read most ovarian cysts, even in post-menopausal women, are benign and don't require intervention. This website by a reputable ob/gyn addresses this at the bottom of the webpage with the heading "Can an ovarian cyst form after menopause."
Absent a genetic predisposition for ovarian cancer such as BRCA1+ or BRCA2+, a woman's lifetime risk of ovarian cancer is less than 2%, 1.3% to be exact from the SEER government statistics here - http://seer.cancer.gov/statfacts/html/ovary.html.
We need our ovaries (and uterus) our whole lives as they have non-reproductive functions including hormone production that has many health benefits. This is borne out by numerous studies. Since the ovaries do continue producing hormones, that explains why cysts can develop after menopause especially in the earlier post-meno years. It also explains a lot of the post-menopausal bleeding episodes that end up being nothing serious...merely some fluctuating hormones. But of course, post-meno bleeding still has to be investigated to make sure it isn't anything serious.
Most ovarian cysts resolve on their own, no intervention needed. Since our parts are so important, if the cysts do get too large or truly look suspicious then you would want to seek out a surgeon who will remove just the cysts (cystectomy) if the cysts are shown to be benign via frozen section.
I was over-treated (organs were removed even though cyst was benign) and my life and health have been negatively impacted in ways I never could have imagined and of which I was not informed by my long-time gynecologist (horrifically rapid aging for one). I have since discovered that hysterectomy and oophorectomy (ovary removal) are two of the most overused surgeries with less than 10% being necessary.
Keep us posted on what you find out. I hope you fall into the majority of these being nothing serious.