Modern Christians look to the Vision of Saint Peter, as depicted in Acts 10 of the New Testament. In the context of a Roman centurion sending messengers to meet with Peter in order to hear about Christianity, Peter is said to have had a vision while praying on his roof, where a sheet full of unclean animals was lowered before him. As Peter rejects the "unclean" foods, a voice supposedly called out to him three times, roughly saying, "what God has cleansed, you shall not call unclean."
In the context of this passage, the accepting of the "unclean meat" is clearly a parallel to the non-Jewish Roman centurion wishing to learn more about the practices of the faith. But this allegory plays into a much broader debate in the early Church as to what of the Jewish religious laws any gentiles wishing to join the Church would have to follow. At this time, Barnabas and Paul have already begun preaching christianity to gentiles in what is now southern Turkey, and the Jerusalem Council was called after reports that Jewish Christians in these areas were insisting the gentile converts become circumcised. In Acts 15, the Council, at Peter's urging, made the decision that gentiles entering into the Church would only have to follow four of the rules of Jewish law, abstaining from fornication and of eating blood, strangled animals, or food offered to idols.
The heavy dietary bent of these four remaining laws that apply to gentiles still points to the importance of this aspect of Jewish practices passed on to its gentile-friendly successor religion. However, the details in Acts 15 on the approach of the Jerusalem Council makes clear their intent to avoid "put[ting] a yoke upon the neck of the [gentile] disciples, which neither our fathers nor we were able to bear." While the law of Moses was still important to the Jerusalem-based early Church, they clearly placed a higher priority on expanding the religion's adoption than on insisting on strict adherence to its rules.
It's difficult to find a non-biased consensus as to when the Acts of the Apostles were written down, but they are often believed to be one of the earliest books of the New Testament to be put to paper after the letters of Paul, with the Acts Seminar at the Westar Institute asserting its writing being around 115 C.E. in Rome. The debates about the rules of the early Church that it documents would have been fresh in the minds of Jewish and gentile Christians in the first century, and increasingly relevant as the religion spread rapidly across the Roman Empire. In a time when the life and teachings of Jesus were still spread by word of mouth, and an emphasis placed on the spreading of the religion, the Vision of Peter could have easily been taken to be a license to forego many of the stricter dietary measures in the laws of Moses, especially by a Roman imperial public for whom such restrictions were foreign.