When I was a child in the 1980s I was captivated by Michael Wood’s documentary In Search of the Trojan War (he also wrote a book with the same name). I had read a fair amount of Greek mythology, prose translations of the Iliad, as well as ancient history. The contrast between the Classical Greeks, the strangeness of their mythology was always something that on the surface of my mind. The reality that Bronze Age Greeks were very different from Classical Greeks resolved this issue to some extent.
Though Classical Greeks were very different from us, to some extent Western civilization began with them, and they are very familiar to us. Rebecca Goldstein’s Plato at the Googleplex was predicated on the thesis that the ancient Greek philosopher had something to tell us, and that if he was alive today he would be a prominent public speaker.
I’m going to dodge the issue of Julian Jaynes’ bicameral mind, and just assert that people of the Bronze Age were fundamentally different from us. And that difference is preserved in aspects of Greek mythology. Though it is fashionable, and correct, to assert that Homer’s world was not that of Mycenaeans, but the 
barbarian period of the Greek Dark Age, it is not entirely true. Homer clearly preserved traditions where citadels such as Mycenae and Pylos were preeminent, and details such as the boar’s tusk helmets are also present in the Iliad.
But aesthetic details or geopolitics are not what struck me about Greek mythology, but events such as the sacrifice of Iphigenia. Like Abraham’s near sacrifice of his son, this plot element strikes moderns as cruel, barbaric, and unthinking. And though the Classical Greeks did not have our conception of human rights, they had turned against human sacrifice (and the Romans suppressed the practice when they conquered the Celts) on the whole, but it seems to have occurred in earlier periods.
The rupture between the world of the Classical Greeks and the strange edifices of Mycenaean Greece were such that scholars were shocked that the Linear B tablets of the Bronze Age were written in Greek when they were finally deciphered. In fact many of the names and deities on these tablets would be familiar to us today; the name Alexander and the goddess Athena are both attested to in Mycenaean tablets.
Preceding the Mycenaeans, who emerge in the period between 1400-1600 BCE, are the Minoans, who seem to have developed organically in the Aegean in the 3rd millennium. This culture had relations with Egypt and the Near East, their own system of writing, and deeply influenced the motifs of the successor Mycenaean Greek civilization. The aesthetic similarities between Mycenaeans and Minoans is one reason that many were surprised that the former were Greek, because the Minoan language was likely not.
Mycenaean civilization seems to have been a highly militarized and stratified society. There is a reason that this is sometimes referred to as the “age of citadels.” Allusions to the Greeks, or Achaeans, in the diplomatic missives of the Egyptians and Hittites suggests that the lords of the Hellenes were reaver kings. In 1177 B.C. Eric Cline repeats the contention that a fair portion of the “sea peoples” who ravaged Egypt in the late Bronze Age were actually Greeks.
So when did these Greeks arrive to the shores of Hellas? In The Coming of the Greeks Robert Drews argued that the Greeks were part of a broader movement of mobile charioteers who toppled antique polities and turned them into their own. The Hittites and Mitanni were two examples of Indo-European ruling elites who took over a much more advanced civilizational superstructure and made it their own. While the Hittites and other Indo-Europeans, such as the Luwians and Armenians, slowly absorbed the non-Indo-European substrate of Anatolia, the Indo-Aryan Mitanni elite were linguistically absorbed by their non-Indo-European Hurrian subjects. Indo-Aryan elements persisted only their names, their gods, and tellingly, in a treatise on training horses for charioteers.
Drews’ thesis is that the Greek language percolated down from the warlords of the citadels and their retinues over the Bronze Age, with the relics who did not speak Greek persisting into the Classical period as the Pelasgians. Set against this is the thesis of Colin Renfrew that Greece was one of the first Indo-European languages, as Indo-European languages began in Anatolia.
The most recent genetic data suggest to me that both theses are likely to be wrong. The data are presented in two preprints, The Population Genomics Of Archaeological Transition In West Iberia and The Genomic History Of Southeastern Europe. The two papers cover lots of different topics. But I want to focus on one aspect: gene flow from steppe populations into Southern Europe.
We know that in the centuries after 2900 BCE there was a massive eruption of individuals from the steppe fringe of Eastern Europe, and Northern Europe from Ireland to to Poland was genetically transformed. Though there was some assimilation of indigenous elements, it looks to be that the majority element in Northern Europe were descended from migrants.
For various reasons this was always less plausible for Southern Europe. The first reason is that Southern Europeans shared a lot of genetic similarities to Sardinians, who resembled Neolithic farmers. Admixture models generally suggested that in the peninsulas of Southern Europe the steppe-like ancestry was the minority component, not the majority, as was the case in Northern Europe.
These data confirm it. The Bronze Age in Portugal saw a shift toward steppe-inflected populations, but it was not a large shift. There seems to have been later gene flow too. But by and large the Iberian populations exhibit some continuity with late Neolithic populations. This is not the case in Northern Europe.
In The Genomic History Of Southeastern Europe the authors note that steppe-like ancestry could be found sporadically during early periods, but that there was a notable increase in the Bronze Age, and later individuals in the Bronze Age had a higher fraction. Nevertheless, by and large it looks as if the steppe-like gene flow in the southerly Balkans (focusing on Bulgarian samples) was modest in comparison to the northern regions of Europe. Unfortunately I do not see any Greece Bronze Age samples, but it seems likely that steppe-like influence came into these groups after they arrived in Bulgaria, which is more northerly.
Down to the present day a non-Indo-European language, Basque, is spoken in Spain. Paleo-Sardinian survived down to the Classical period, and it too was not Indo-European. Similarly, non-Indo-European Pelasgian communities continued down to the period of city-states in Greece.
These long periods of coexistence point to the demographic equality (or even superiority) of the non-Indo-European populations. The dry climate of the Mediterranean peninsulas are not as suitable for cattle based agro-pastoralism. This may have limited the spread and dominance of Indo-Europeans. Additionally, the Mediterranean peninsulas were likely touched by Indo-European migrations relatively late. Much of the early zeal for expansion may have already dissipated by them. The high frequency of likely Indo-European R1b lineages among the Basques is curious, and may point to the spreading of male patronization networks, and their assimilation into non-Indo-European substrates where necessary. R1b is also found in Sardinia, and in high frequencies in much of Italy.
The interaction and synthesis between native and newcomer was likely intensive in the Mediterranean. For example, of the gods of the Greek pantheon only Zeus is indubitably of Indo-European origin. Some, such as Artemis, have clear Near Eastern antecedents. But other Greek gods may come down from the pre-Greek inhabitants of what became Greece.
Ultimately these copious interactions and transformations should not be a great surprise. The sunny lands of the Mediterranean attracted Northern European tribes during Classical antiquity. The Cimbri invasion of Italy, Galatians in Thrace and Anatolia, the folk wandering of Vandals and Goths into Iberia, are all instances of population movements southward. These likely moved the needle ever so slightly toward convergence between Northern and Southern Europe in terms of genetic content.
In relation to the more general spread of Indo-Europeans, I believe there are a few areas like Northern Europe, where replacement was preponderant (e.g., the Tarim basin). But I also believe there were many more which presented a Southern European model of synthesis and accommodation.