What you are describing is the prevailing attitude among most people who call themselves Christian. The general approach is to consider the Bible to be two totally different segments—the Old Testament that has a harsh God (the Father) who dealt with people one way and the New Testament, which has a loving God (Jesus the Son) who deals with people in a totally new and different manner. Many problems come from such an artificial dissection of God's Word.
Some want to argue that Jesus only kept the Commandments and upheld them because He came as a Jew and had to keep them until He died. They then claim that the apostle Paul did away with the Commandments of God and only focused on love and mercy. However, Paul told the Corinthian church that the Commandments were what really mattered in life (1 Corinthians 7:19). This was written around A.D. 55, which is 24 years after Christ died. That doesn't sound like the Commandments ended at His death! In addition, somewhere around A.D. 90 the apostle John wrote that the way we know we love God is if we keep His Commandments, and if we don't keep them, we don't have God's love in us (1 John 2:3-6). You might also notice this passage ends with the statement that we should walk as Jesus walked—meaning, live like He did.
It's obvious that the law that is to be "written in our hearts" is the law of God, summarized by the Ten Commandments. If you wish to read more about them you can read our booklet on the Ten Commandments.