The Mycenaeans were eventually succeeded by the Dorians, a group of Greek-speaking tribes who invaded the Greek mainland from the north around 1100 BCE. The Dorians brought with them a new dialect of Greek, which eventually became the dominant form of the language. The classical Greek civilization, which flourished from around 500 to 300 BCE, was dominated by the Dorians and their Greek-speaking descendants.
So, while the Mycenaeans were the first to speak Greek, the Dorians are generally considered to be the first truly Greek people, as they were the ones who shaped the culture and language of classical Greece.