Actually polygamy never stopped. It was practiced by the Jews until the 11th Century (if memory serves me correctly). It was practiced in both Old and New Testament times. And it is still practiced around the world in various cultures, especially in the Middle East. Polyandry (a woman with multiple husbands) is much rarer but is also practiced in a culture in Nepal/Bhutan.
There was a rabbinical debate concerning polygamy during the time of Christ. Supporters of polygamy pointed out that the patriarchs were polygamous, that the Law does not forbid polygamy, and in fact the Law actually regulates specific polygamous relationships thus indirectly allowing polygamy.
Rabbinical opponents of polygamy pointed to the flood and especially to creation. When speaking of creation they would quote Gen.2 from the Septuagint, "and the TWO shall become one flesh." They quoted the Septuagint because the Hebrew scripture actually says, "and THEY shall become one flesh".
It is significant to note that Jesus when speaking with the Pharisees concerning the "Any Matter" divorce debate actually quotes from the LXX saying "and the TWO shall become one flesh." Thus it can be said that Jesus endorsed monogamy as part of the divine ideal of marriage, the best, what God originally intended. However, one can not say that Jesus legislated monogamy, or implied that monogamy was the only viable family structure.
Since the fall of mankind, various cultures have found polygamy to be helpful, even necessary, particularly when war killed off many of the men. Not only that, but men naturally do not loose the biological drive/desire to procreate until very late in life, if ever. Whereas women loose the biological drive/desire to procreate mid-life. (Of course I'm speaking in broad generalities.)
So if Jesus endorsed monogamy, but did not legislate monogamy much less forbid polygamy, where did the church pick up the doctrine of monogamy? It came predominantly from the churches Greco-Roman "heritage."
The first great division in the church was between Jews and Gentiles. Jewish Christians were persecuted by the Romans for being Jewish, and by the Jews for being Christians. And thus by 200 A.D. the Jewish branch of the church had been persecuted almost from existence, whereas the Gentile branch flourished. Largely due to this persecution, the Gentile branch increasingly and quickly disassociated itself from Judaism, including changing the day of worship and forsaking completely the Law. By 400 A.D. the church had fully disassociated itself from anything Jewish.
Anyhow, this great divide left the church relying very heavily upon it's own culture, the Greco-Roman culture to draw from for it's values and philosophies. One area that it drew from most was the family. Platonic thought permeated and still permeates much of the Western church. Concerning family, marriage, and sex, Plato, Greco-Roman thought was pretty messed up. In the early Gentile church, sex actually came to be looked on as evil even within marriage. And monastic life became the preferred way of life. The seeds of this can be seen as early as in 1 Cor. when Paul addresses the problem of Christians prefering singleness to the point of even divorceing their spouses if necessary.
Anyhow, the traditional doctrine of MDR (including the legislation concerning monogamy) arose out of a combination of these errant attitudes and beliefs concerning family, marriage, and sex of the early Gentile church; and a misunderstanding of what Jesus said concerning MDR.
Jesus endorsed monogamy, but He didn't forbid polygamy. And the Law regulated polygamy, but it too did not forbid polygamy.
Please don't misunderstand me, I too endorse monogamy as the best and healthiest family structure. But I would not expect an Arab with 4 wives to divorce 3 of his wives in order to become a Christian. And if I lived in a country/culture that allowed polygamy, I might consider polygamy (if my wife would let me!)
Blessings,
Sherman