So what is righteousness? Short answer: not wrongishness.
Righteous means: the way it ought to be. As Christians we are rightly amazed at the way Jesus lived righteousness (the way it ought to be) and then gave that to us by grace through faith. We should never lose our awe at God as the one who is by definition righteous and the one who describes righteousness in his written and living word. For the sermon on the mount, Jesus is using righteous in its most practical form. Try it:
Blessed (on track, getting it right) are those who are persecuted (treated badly by those who aren’t right) because they are so thoroughly righteous (living the way they ought to live). Why? Because they are the ones who really are living AS IF God was in charge of right and wrong now and at the conclusion of history.
How would you paraphrase what Jesus said to someone without a lot of Jewish understanding?