The main things to work with are: comparison with related languages (e.g. Arabic, Hebrew, Ethiopic); sources in which Babylonian and Assyrian words are written in the Greek or Aramaic alphabets; sources in which Greek and other foreign words are written in Babylonian and Assyrian script; patterns within the spellings of Babylonian and Assyrian words.
Much about their pronunciations is still being discovered, though for obvious reasons much else is lost forever.