In the end you have a mess (see: same sex marriages today) where you think people should know better and realize this isn't about what's "right or wrong" to any individual on any level, but a matter of discrimination when it comes to benefits that are not being given to one group but to another. But they don't always know better.
In the US, we're barely scraping by trying to get the same benefits to gay couples, let alone any mix of more than two people. A movement to do what you suggest...I can't imagine what kind of uproar there will be and how many people will be conflating polyamory with polygamy and being very against it as a result. At the minimum.
Geeks: Polyamory? -> {Manuel Garcia O'Kelly and his extended family, Moon Colonies, Heinlein, good associations}
Non-Geeks: Polyamory? -> {Hippy communes, commun...ism, cold war, evil, bad associations}
Yes. Because that's what those Russians were doing on the other side of the Iron Curtain... having massive orgies.
Well that does put a slightly different spin on "From each according to his ability, to each according to his need".
A lot of people have the idea that any type of polygamy is inherently sexist, and while that's true for traditional polygamy, it's also true for "traditional" monogamy. The historic Mormon polygamy that the fundamentalists get arrested for practicing was indeed sexist, but so were the 19th century monogamous marriages practiced by most other Americans. Contemporary monogamous marriages aren't nearly as sexist, and there's no reason contemporary polygamous arrangements--i.e. polyamory--are necessarily sexist, either.
Polyamory (and the various "open relationship"/non-monogamous lifestyles related to it) are less tied to the ritual of marriage and are often more equitable to women than traditional polygamy. I actually know some polyamorous people who live under the same roof and make babies, for instance. But some people just choose not to mind that much when their partners have sex with someone else, as long as it's safe and consensual. As I said, there's no set rules, other than it has to be safe, sane, and consensual for all parties involved. Which is worlds away from fundamentalist nutjobs in Utah marrying dozens of 13 year old girls.
What if people want to have multiple marriage partners? Wouldn't it be polygamy then?
For legal reasons, most polyamorous people separate their lifestyle from the legal institution of marriage. As another post implied, poly people are a few decades behind the gays in terms of social acceptance--it's not likely the marriage laws will be fixed enough for poly people to use them in the near future. (Lots of poly people are married, of course, but only to one person.)
"Polyamory" is kind of a neologism anyway. Kind of like when the gays decided to be "gay" instead of "homosexual". Polyamorous people are in fact polygamous in the sexual and romantic sense, but not in the technical sense of marriage.
In simple terms, Polyamory is the practice of open relationships, where Polygamy is the practice of having multiple married partners.
[1] cf. http://www.paulgraham.com/say.html