I have chatted with Christians on another web board long ago. I asked a few about polyamory. Most said that God wanted us to be monogamous. However, the Old Testament is full of laws supporting polygyny. There was a diversity of opinion about this.
Some people thought that this was one valid model of marriage that God approved of, but no lesbian action nor polyandry.
Some thought that God tolerated it like he tolerates divorce. However, there are passages where God helps a man marry more than one wife. Some people rebutted by saying that God wanted man and women to be "one flesh." However, this didn;t work because there is a passage in the Bible where a man is "one flesh" with a prostitute.
Some thought that God allowed for this bad for of marriage in the Old Testament, but Jesus died to remove old rules like this. The problem with this was that the Old Laws were called perfect and eternal.
So I think going with the Old Testament didn't resolve anything for the people who take the whole Bible literally. The more liberal Christians tended to go with more of a New Testament view of loving your neighbor as yourself. So if you focus on the love side of it, polyamory should fit in nicely.