Safe to say, the Islamic establishment would be rather ambivalent about uttering the word polyamory—there is a bit of religious “baggage” there if I may be so concise. For those who are squeamish, the idea of having multiple partners (and perhaps worse, the possibility of “feelings” being shared between more-than-two, for most people, is unavoidably offensive.
In a funny way, Islam is progressive in the arena of “free love”. This is because Islam acknowledges the capacity for people to love more than one person through its jurisprudence. The two key drawbacks, of course, are that this privilege is only available for men and by just one of two methods: marriage or slavery. A Muslim man has access to up to four wives, as well as any “human property”.
Removing tongue from the innermost depths of cheek, I will say that Islam is not so progressive as I might have misled the more gullible of us to believe. In any case, Islam’s sanction of polygamy led me to the following cartoon.