As you've guessed, the Neo-Assyrian and Neo-Babylonian periods were predated by the Old and Middle Assyrian periods and the Old and Middle Babylonian periods.

The Neo-Sumerian period, more often called the Ur III period or Third Dynasty of Ur, came after the Early Dynastic period (the floruit of the Sumerian city-states), with the Akkadian period sandwiched in between. The term Neo-Sumerian has fallen out of favor in recent years; it has become clear that Mesopotamian society in the Akkadian and Ur III periods was a complex mishmash of Semitic and Sumerian cultural elements, so Neo-Sumerian is not a very accurate or useful label. Akkadian continued to replace Sumerian as a spoken language during the Ur III period, and Sumerian was probably a dead language by the end of the period, although historians don't agree on exactly when Sumerian died out as a spoken language.

I discussed the chronology of Mesopotamia, with links to my earlier responses on the topic, in a recent AskHistorians post.