"旧石器时代的" can be translated into English as "Paleolithic" or "Paleolithic Age/Period" (when referring to the specific time period).
Paleolithic is the most direct and commonly used term in academic and general contexts to describe the era characterized by the use of simple stone tools.
If you want to specify it as a time period, you could say "the Paleolithic Age" or "the Paleolithic Period."
For example:
"Paleolithic tools" (旧石器时代的工具)
"The Paleolithic Age was a crucial period in human prehistory." (旧石器时代是人类史前史上的一个关键时期。)