Previously on Paul.
Most scholars would also recognize today that while Luther's sense of "initial justification" was mostly accurate to Paul, his sense of what the phrase "the righteousness of God" meant in Romans 1:17 was probably wrong. The Roman Catholic Church took it to mean the justice God distributed toward believer and non-believer alike. Luther took it to be a "legal fiction," God considering us righteous even though we are not, a righteousness from God.
But the Jewish background to the idea of the righteousness of God points in another direction. For example, Psalm 98:2 says that "The LORD has made his salvation known and revealed his righteousness to the nations." See how much this verse has in common with Romans 1:17?  It mentions God's righteousness and speaks of it being revealed. It also has a sense of this revelation going out to all the nations, just as Paul understood the gospel to be for the Gentiles as well as Jews.  The verse does not speak of us becoming righteous, but of God's righteousness as he brings about the salvation of Israel.
This psalm is not the only place where God's "righteousness" and his "salvation" are mentioned parallel to one another.  The second half of Isaiah (chaps. 40-66) was of great significance to the earliest Christians, and Paul himself occasionally alludes to these chapters (e.g., Rom. 15:21). These sorts of parallels between God's righteousness and the salvation he is bringing permeate them. The Greek version of Isaiah 51:5 Paul used says, "My righteousness draws near quickly, my salvation will come out like a light. The Gentiles will hope on my right arm." 
With such a consistent Old Testament background--also found in some of the Jewish literature of Paul's own day --Paul almost certainly inherited a sense that the expression, "the righteousness of God" referred to God's righteousness as God's propensity to save Israel. Paul of course did not have to stop with what the phrase had meant before him, but it is pretty clear what his starting point was.
So when Paul says that "the righteousness of God is revealed in the gospel" in Romans 1:17, he is talking about God's relationship with his people and in particular God's propensity to save his people, God's "saving righteousness." Righteousness is a relational term in Jewish thought. It is not about some abstract quality God has but about a specific way that God relates to his people and the world. It is a word that can refer to God's justice, as we will see in Romans 1:18, where Paul goes on to talk about the wrath of God toward ungodliness. But in Romans 1:17 Paul is talking about good news, how God has acted through Christ to bring salvation to his people, which now includes not only believing Jews, but believing Gentiles as well.
So Paul starts with the way Jews already thought of the righteousness of God, God's faithfulness to Israel both in judgment and salvation. Then he Christianizes the concept. Now it is only through Christ that this righteousness comes to God's people. And now God's people is not exactly the same as Israel but only includes believing Jews and is extended to include believing Gentiles as well.
And just maybe, Paul exploits the ambiguity of the phrase "the righteousness of God" so that when he gets to Romans 3:21, he not only alludes to God's righteousness. Maybe Luther was not so far off here as we think. When Paul says, "now, apart from law, righteousness of God has been disclosed,"  maybe he intended the Romans to hear a double meaning, a double entendre. Not only has the righteousness of God" been revealed in a new way through Christ, God's righteousness. But maybe he means his readers also to hear that a "righteousness" is now available for us "of God," that is, "from God."
 We should add here to our list of key books on Paul, Richard Hays' celebrated Echoes of Scripture in the Letters of Paul (New Haven, CT: Yale University, 1989). In this book, Hays explores passages in Paul that may "echo" passages from the Old Testament. An echo is something a little weaker than a full blown allusion, where it is clear an author quite intentionally is pointing toward a specific Old Testament (or other) text.
 The next verse reads, "He has remembered his love and his faithfulness to the house of Israel; all the ends of the earth have seen the salvation of our God."
 Many will know that Hebrew poetry does not rhyme sounds, but ideas. The verses we are discussing here use "synonymous parallelism" where one line is followed by another line that roughly says the same thing.
 My translation.
 The Dead Sea Scrolls, for example, think of God's righteousness in similar terms. For example, "if I stumble, the mercies of God shall be my salvation always; and if I fall in the sin of the flesh, in the justice of God... will my judgment be... he will judge me in the justice of his truth, and in his plentiful goodness always atone for my sins; in his justice he will cleanse me from the uncleanness of the human being" (1QS11.12). Translation from Florentino G. Martínez and Eibert J. C. Tigchelar, The Dead Sea Scrolls: Study Edition, vol. 1 (Grand Rapids: Eerdmans, 1997) 99.
 My translation.