They slowly got replaced by Akkadians around 2300 BC. Ironically you play as Sumerians in the Sargon of Akkad campaign in RoR, which is technically wrong. Since the Bronze Age started like 3300 BC in the Middle East they were clearly still in the Bronze Age.
But yeah, very much before the Iron Age that started like 1200 BC in the Middle East.
I think it would be really cool to have a Bronze Age Spin-off idea.