It has been said that the New Testament is in the Old Testament - concealed - and that the Old Testament is in the New Testament - revealed.
When Christians say they don't follow the Old Testament they should mean that they are not bound by sacrificial and ceremonial and dietary laws which were symbolic types and foreshadows of what God was going to fulfill through Jesus Christ and the inner purity that He was going to create when a person is born again by the indwelling presence of his Holy Spirit.
Jesus said "Do not think that I came to destroy the Law or the Prophets. I did not come to destroy but to fulfill. Matthew 5:17
This is explained in the epistles.
Colossians 2:11-17 "In him you were also circumcised, in the putting off of the sinful nature, not with a circumcision done by the hands of men but with the circumcision done by Christ, having been buried with him in baptism and raised with him through your faith in the power of God, who raised him from the dead. When you were dead in your sins and in the uncircumcision of your sinful nature, God made you alive with Christ.
He forgave us all our sins, having canceled the written code, with its regulations, that was against us and that stood opposed to us; he took it away, nailing it to the cross. And having disarmed the powers and authorities, he made a public spectacle of them, triumphing over them by the cross.
Therefore do not let anyone judge you by what you eat or drink, or with regard to a religious festival, a New Moon celebration or a Sabbath day. These are a shadow of the things that were to come; the reality, however, is found in Christ.
Hebrews 10:1 The law is only a shadow of the good things that are coming--not the realities themselves. For this reason it can never, by the same sacrifices repeated endlessly year after year, make perfect those who draw near to worship....11 Day after day every priest stands and performs his religious duties; again and again he offers the same sacrifices, which can never take away sins. 12 But when this priest had offered for all time one sacrifice for sins, he sat down at the right hand of God. 13 Since that time he waits for his enemies to be made his footstool, 14 because by one sacrifice he has made perfect forever those who are being made holy.