There weren't any "Rosetta Stones" for cuneiform, right? So how did we figure out how to read these languages?
There as a kind of “Rosetta Stone” for cuneiform. It is called the Behistun Inscription (pic), but it wasn't as easy to use as the Rosetta Stone for a couple of reasons. It is a rock carving in the side of a mountain between Iraq and Iran that the ancient Persians scraped away the walkway from, making it exceedingly difficult to closely approach. Making matters worse, the three languages it contains are all written in cuneiform characters, so it took a fair amount of work to be able to recognize that one of the languages was related to a language we already knew.
The story of the decipherment is well publicized, and you can read the full account in a variety of sources. I list some below.
The condensed version is that a British scholar, Sir Henry Rawlinson, was able to make a plaster cast of the inscription and publish hand drawn copies of it in 1847. He deciphered the Old Persian portion, with the help of some other recent discoveries, because Old Persian is related to the Persian languages known today, such as “Farsi.” British scholars then set to working on the Akkadian and Elamite portions, which contain nearly identical translations of the Old Persian section. It took them about ten years, but they were able to decipher them successfully with the help of recent finds that had come out of Nineveh. In 1857, the Royal Asiatic Society had four scholars independently translate the same Akkadian text, and when they all came up with the same translation the language was considered deciphered.
The decipherment of Sumerian is a much longer and more technical story that makes for boring reading. It is necessary to read some Sumerian words in order to be able to fully read most Akkadian texts, so Sumerian was partially deciphered along with Akkadian, but gaining the ability to read purely Sumerian texts took much more work. Sumerian is a language isolate, meaning it is not related to any other language known. We rely on Akkadian linguistic texts (dictionaries and commentaries) to make sense of the difficult parts, and it took scholars about 90 years to have a handle on most of the difficulties. Even today, major debates rage about the precise meaning of the verbal system, and translations can vary considerably.
Here is some references for the full stories:
