Jews don't sacrifice animals because their temple was destroyed. In the days of the patriarchs, animals were sacrificed at high places. Before the Israelites established the temple, they sacrificed at the tabernacle. Once the temple was built, worship was centralized there. The high places, according to the Biblical narrative, were torn down. This was all codified in the Law, specifically Deuteronomy. And in the books of Kings and Chronicles, the kings of Israel are judged as either right or wrong in the eyes of the Lord primarily based on how faithfully Israel worshiped Yahweh under their rule. Once the temple was destroyed, there was no place to conduct sacrifices that are acceptable in God's eyes.
And with respect to the Law, I don't think it's as cut and dry as some Christians make it out to be. Surely you still have to keep the 10 commandments, for example. I've heard it argued that the moral laws are to be kept but the ritualistic laws are to be done away with. But I'm not sure how well that sort of argument can work when we're supposing that morality is rooted in God and His preferences.