You were being taught essentially nothing about either period, because the Bronze Age (and anything earlier) as an historical period was virtually unknown in antiquity. Let's start with Greece, then Rome.

The visible ruins of the Mycenaean past were variously assigned to elements of the "Age of Myth" by those who came to live in "Dark Age" Greece after the collapse. For instance, the Greeks of the 8th century could walk in the ruins of Pylos and talk about Nestor (of Iliad fame) ruling there, or sail to their modern-day Troy (quite a small little place) and talk about the great war that had happened there in the distant past. Their connection to the "real" Mycenaean civilization was wholly absent, though the language they spoke and began to write in the 8th century was directly related to Mycenaean Linear B.

Though some will argue against me here, in my opinion it is not even possible to clearly demonstrate what we archaeologists call "continuity of cult" at ritual sites between the end of the Bronze Age and the 8th century. Yes, there are some exceptions which seem to challenge this assumption (like the ash altar at Olympia, which stretches back to the 10th century at least), but for the most part, places that were sacred in Mycenaean times almost always experienced a clear and abrupt break in ritual practice. Sometimes cult resumes there, centuries later, and sometimes it does not. The Mycenaeans had often buried their dead in tholos tombs, sometimes called beehive tombs. Beginning in the 8th century (a busy time, yes), Greeks began to engage in cult practice at these tombs. It is clear they understood that people of an earlier time were/had been buried there. Sometimes, they assigned these structures to this or that Homeric hero; at other times, particularly in Attica, they began to deposit offerings in the entrance of the tombs in order to co-opt the unknown deceased as their ancestors. This false claim of kinship reinforced their right to occupy the land, and to "prove" their claim of autochthonia, or the idea that they and their ancestors had always occupied that land. This phenomenon is called tomb cult. That this past could be co-opted in this way is, for me, very clear evidence that there was no knowledge of the "truth" of the Mycenaean civilization.

Now to Rome. The story is much the same. The Terramare culture of the Italian Bronze Age thrived and disappeared around 1200 BCE, and there is little evidence that the Romans ever encountered or in some way remembered its existence. Dionysus of Halicarnassus, writing in the early principate (so over a thousand years later) mentions the legendary Pelasgians as living in the Po Valley in the generations preceding the Trojan War. This may indicate some kind of collective memory of Bronze Age occupation, but color me skeptical. It is the same for the Villanovan culture which followed. Apparently an off-shoot of the Urnfield or Halstatt culture of Europe, they were around northern and central Italy until about 700 BCE, when there was a relatively seamless transition to the Etruscans. Roman myth's earliest memories belong here -- the legendary founding of the city by Romulus and Remus was in 753 BCE. Alternately, the Aeneas foundation myth belongs to the Trojan cycle; the age-peers of Aeneas were roughly Telemachos son of Odysseus, and Neoptolemos son of Achilles.

It is possible that the "Hut of Romulus" in Rome was, originally, some kind of Iron Age Villanovan ruin which was later interpreted by the Romans as the house of their legendary founder. If so, the phenomenon is very similar to that happening at the Mycenaean tombs as they were identified as the resting places of Homeric heroes.

A question similar to yours has been posted here before, and got a great answer from /u/xiaorobear which details some of the (Greek) literary evidence.