Why were animal sacrifices required for those in the old testament period (but not required of us); if jesus was destined to die for all our sins?

Genesis 4:3-4; Leviticus 3:2; Leviticus 16:5; Ezekiel 44:1;Numbers 6:10-11; and Exodus 29:10-14 are some of the many places in the Bible that speak of God requiring animal sacrifices or sacrifices being made to God for expiation of sins or for thanks. If God's plan all along was that Jesus would come as a perfect sacrifice to atone for the sins of all of humanity; why were animal sacrifices required of the people of those "Old Testament" times, but not of our time; yet Jesus was destined to die for the sins of both ("groups")? My question is not restricted to the purpose of animal sacrifice as asked in this post: (https://christianity.stackexchange.com/questions/842/what-was-the-purpose-to-animal-sacrifices) ; but I ask why we no longer have to offer them when they were required of earlier generations despite the fact Jesus would die for their sins as well as our sins.