Searching for dikaioma in the Septuagint reveals that it is used 122 times (see the appendix). Examining those passages and their context shows that the righteous acts begin in the book of Genesis and are mentioned throughout Scripture all the way to the final Revelation passage. Since the Septuagint is a translation (albeit an excellent one), we cannot be dogmatic about every instance where dikaioma is used. The context of passages using the word is enlightening, however.
Among the Septuagint's verses that use dikaioma, one passage of Scripture informs us as to a purpose of the Law of Moses: the revelation of G-d's righteousness.
"See, I have taught you statutes [dikaioma] and judgments just as the LORD my God commanded me, that you should do thus in the land where you are entering to possess it. So keep and do them, for that is your wisdom and your understanding in the sight of the peoples who will hear all these statutes [dikaioma] and say, 'Surely this great nation is a wise and understanding people.' For what great nation is there that has a god so near to it as is the LORD our God whenever we call on Him? Or what great nation is there that has statutes [dikaioma] and judgments as righteous as this whole law which I am setting before you today? (Deuteronomy 4:6-8)
Throughout the Septuagint dikaioma is used in place of the following Hebrew words:
חק chok (Strong's #2706) - a royal appointment/enactment
חקּה chukkah (Strong's #2708) - something prescribed, an enactment, a statute
משׁפּט mishpat (Strong's #4941) - a judgment, verdict, or ruling
דּרך derek (Strong's #1870) - a way (used in Job 34:27 speaking of G-d's ways)
פּקּוּד pikkud (Strong's #6490) - a precept, an order
To the writers of the Septuagint, it appears that many of the royal enactments, statutes, judgments, and orders of G-d are "righteous acts."
Let's put it all together and summarize.