One interesting thing about the Iliad is that there is a chapter called the catalog of the ships. In this chapter is a big list rulers and their hometowns joining Agamemnon's attack on Troy. What historians have found out is that the list follows the Mycenaean settlement patterns rather than classical ones. Some cities had fallen out of use, etc. And of course the Iliad was very popular in classical Greece.
To an extent-most of what we know of classical Greece are their oral myths, which would have been the Iliad and the Odyssey. While there was, and still is, some contention about the historical validity of the accounts, when taken metaphorically, we can see them for being a rough history of Greece.
There's observable astronomical phenomena in the Odyssey when Odysseus returns home from his ten-year voyage (compounded by ten years at Troy) that pinpoints the date of his return as April 16, 1178. This would place the Iliad as occurring sometime in the roughly twenty years prior to this (sometime around ~1200 BCE, if it happened). Odysseus' return voyage takes him ten years and he is beset the entire way by difficulties, as well as raiding a port to begin with.
I took an Ancient Greece history class last quarter, and this is strongly condensing it, but in essence, the Odyssey's troubles can be seen at metaphorical for what one would expect to be going on in Greece (Achaea) at the time. There was competition with the Hittites, I want to say, and in addition to that, difficulties with pirates and marauders. The trade that had evolved to connect mainland Greece with the Minoans and Mycenaeans as well as the Egyptians collapsed in the years leading up to this point and there is a theory that by the year 1177, Classical Greek society was FUBAR.
If you meant the Mycenaeans and Classical Greeks that lived around the same time, yes, there is evidence of trade between them as well, I believe primarily pottery and frescoes. (That may have been the Minoans on the frescoes though).
tl;dr: A little, arguably, in the form of their myths and legends, which retell a history of Greece that is currently being excavated.
Sources: Cynthia Shelmerdine's book on Classical Greece
Carol Thomas, Citadel to City State
Homer's The Odyssey (Fagles translation, my preferred)
E.H. Cline, 1177 the Year Civilization Collapsed
Edit: made sources readable