First off, this word "righteousness" has dropped out of people's consciousness. It is no longer spoken about.
Second, this word also means Justice. Dikaios is either translated as Righteousness or Justice. And what passes for the meaning of Justice today---DOES NOT match what the Ancients meant by this word.
Third, this word is gargantuan. It has a ton of meanings and is very varied.
I want to discuss every single ramification of the meaning of this word Dike, Dikaios, Dikaiosyne.
But first in this post, I want to discuss something about what I found in C. M. Bowra C. M. Bowra in The Greek Experience writes:
Quote: "The word dike which we translate 'justice', seems to be derived from the boundaries of a man's land and conveys metaphorically the notion that he should keep within his own sphere and respect that of his neighbor". pg 99
I read somewhere that "dike" was the Boundary stones of a man's property? Is this right? Now is this the meaning of Dike in the Ionian sense or in the Doric sense? Is Ionian Greek fundamentally different from Doric Greek?
I am very intrigued by this meaning of Dike as a boundary. Does this also go into Dikaiosyne? Is a part of righteousness/justice about respecting boundaries? "To stay within the Lines"?
In an article by Edward Goldsmith, "Archaic Societies and Cosmic Order — A Summary", he has this to say: Across the world, from the beginnings of prehistory, the belief that society must follow a certain path -- or 'Way' -- in order to maintain itself, and the wholeness of the world around it, has been a common theme running through many societies and cultures. This Way, which a society must follow in order to maintain the order of the cosmos, is defined as that which conforms to traditional rules, or 'laws' -- laws which the Ancient Greek referred to as the Nomos, or the Dike -- meaning justice, righteousness or morality. The Dike was "the way of the World, the way things happen." 
 1.) Jane Harrison, Themis: A study of the Social Origins of Greek Religion, Cambridge University Press, 1927, p.517.
Dike is seen in the Cosmos. All things have Dike--They follow a law, their prescribed paths and they don't do anything else. They do one job and they stick to their paths. Dike is the way things work.
In W.K.C. Guthrie's work, "The Greek Philosophers" he says that Dike is justice, Dikaios is an adjective meaning "just", and Dikaiosyne "The state of being dikaios". (pg 6).
At what point do we translate it as "righteousness"?
Dear Wheeler, forgive me if I sound like a 'teacher', but I believe you make a serious mistake. Articles and studies, what in scholarship is rightly called secondary literature, should have also a secondary position in your thinking. Don't start with that. Forget about Harrison, Guthrie, etc., and start with the start, i.e., with the sources. Search for justice in Homer, in Plato and the Greek Fathers. These are the main sources on your current subject. Form your questions and doubts in the studying of the sources, then study again the sources, and only after that allow yourself to spend some time with secondary readings.
To your current question. I can't understand what perplexes you. Can't "Right" and "Just" be synonyms?
Well for one thing, Righteousness is a Virtue. In the psuedo-Aristotlian work, "Virtues and Vices", it says, "To righteousness it belongs to be ready to distribute according to desert, and to preserve ancestral customs and institutions and the established laws and to tell the truth when interest is at stake, and to keep agreements. First, among the claims of righteousness are our duties to the gods, then our duties to the spirits, then those to country and parents, then those departed; and among these claims is piety, which is either a part of righteousness or a concomitant of it. Righteousness is also accompanied by holiness and truth and loyalty and hatred of wickedness." v. 2.
It says to 'preserve' and then to "keep agreements". This also means to be lawabiding doesn't it? Does righteousness mean to be a traditionalist? I found a corallary in the Book of Genesis where one of the commands God gave to Adam was "to Cultivate and to Keep" Keep being synomous with "to preserve".
I am not schooled in Greek, nor do I sit at a college library, nor do I make a million dollars to purchase books, nor do I have the time since I am not an academic but have a regular manual labor job. So the demand I go to "original sources" makes it difficult for me to undertake. I wouldn't be here if I did.