I think I first saw reference to Martin P. Nilsson's "The Mycenaean Origin of Greek Mythology" in the same bibliography where I learned of Denys Page's "History and the Homeric Iliad." The two books cover some of the same ground, but Nilsson's scope is broader than just the Iliad or Homeric epics generally. He carefully examines the traditions of Greek mythology to determine what aspects must go back to at least the Mycenaean era in which Homer's poems are set. Incidentally, it should be noted that Nilsson is primarily concerned with the tales of legendary heroes like Heracles, Theseus, Oedipus, and the warriors of the Trojan war, et al., versus the stories told about Zeus, Hera, and the other Olympian deities. Only in the last chapter does he consider the latter group, and then only to argue that the idea of Zeus as the monarch among gods, with a council surrounding him on Mount Olympus, also must date to the Mycenaean era.
How does Nilsson determine whether particular legends are of more ancient or recent origin? He looks for details that cannot have been current in the era of Homer and later Greeks. For instance, if a particular legend is set in places like Mycenae, Pylos, or Tyrins, which were quite powerful in the Mycenaean era but rather insignificant in later times, then that is a strong case that the legend itself must date from the Mycenaean era. Another technique is to look at the names of the people involved. If the names can be easily derived from Greek words, this suggests a later date; names like Atreus, Odysseus, or Achilleus seem to derive from some non-Greek source.
"The Mycenaean Origin of Greek Mythology" dates from 1930, before the Mycenaean writing system Linear B was deciphered. But I don't necessarily think the translated tablets would have added much to Nilsson's arguments. He makes good use of the archaeological discoveries that had happened up to his time, and compares the evolution and development of myths and legends from a more recent period (for instance, the German Nibelunglied, which incorporates some details whose historical origins are known) to make his case for the origins of Greek myths. I thoroughly enjoyed reading this book, and would recommend it to anyone interested in the subject matter.