My own 'coming out' with family & friends was mixed, friends being wonderful, family being a big mess. In hindsight, I really don't regret telling the family that now disapproves. I only wish I had better prepared myself for the backlash.
As for advice, here's my two cents. If you have shared difficult news with your family before, sometimes you can use their reaction then to gauge their reaction to poly now. Have you come out as gay, gotten divorced, followed a different faith, revealed anything that they disapproved of? Based on their past reactions, you may be able to prepare yourself.
And as unfair as it may be, you may need to prepare for a strong negative reaction on their part and accept that risk. I was inspired to be honest with the people in my life because I was happy, unashamed, and it just didn't feel right to hide anyone that I loved so much. I expected questions, concern, anger, but I believed that our family dynamic could support that. In my case, that was naive based on my family history.
As a result, I'm estranged from my Dad and that side of my family. My attempts to reach out have been ignored and I continue struggling to let things be for now, hoping he will come around while counting the months that are turning into years.
However, I have also discovered a true family in the form of our friends. They have truly been the most incredible people in our lives and their love and support have been unconditional, I struggle for more words to describe how much they have meant. They are our family.
"There are worse crimes than burning books. One of them is not reading them." - Ray Bradbury