Recent studies of apes suggest that our language didn't evolve from grunts, but from gestures. Chimps and bonobos use a complicated set of gestures to communicate, showing an understanding of context, and even developing regional dialects. Ed Yong has a nice summary:
"All primates can communicate with each other through facial expressions, body postures and calls, but humans and apes are unique in their use of gestures. These go beyond simple postures or walking patterns - they are movements of the hand, limbs and feet, specifically directed at another individual.
We think of language as mainly spoken or written but gestures play an enormous, often overlooked role. After all, isn't a speaker who waves their hands animatedly more engaging than one who stands motionless behind a podium? Gestures are such an intrinsic part of the way we communicate that a blind speaker will naturally make them even when speaking to a blind audience. And babies use gestures long before they learn their first words.
Chimps and bonobos also differed considerably in their vocabulary of gestures, with each species having its own 'gesture culture'. The two groups of bonobos even used slightly different sets of gestures to each other."