People already do follow their own moral codes. However, morality is subjective. What's "right" to you isn't necessarily "right" to someone else. Additionally,

People already do follow their own moral codes. However, morality is subjective. What's "right" to you isn't necessarily "right" to someone else. Additionally, it's part of human nature to act in our own self-interest. Call it antisocial if you want, but it's part of us. We're people, not ants. Trying to fight against that desire is a losing battle. Instead, the focus should be entirely on the system itself to create such tremendous disincentives for crime/cheating/etc that it's in our self interest to follow the law (e.g. the laws against theft and murder that you pointed out).