There are no really reliable estimates for the population of Mycenaean Greece, although scholars have supplied some (more or less educated) guesses. On the more conservative end of the scale, Stanford's Mitsotakis professor Josiah Ober has written that:
The population of Hellas in the Mycenaen period (including Thessaly and Crete) was somewhere in the range of 600,000 people.1
At the opposite end, the British historian and Stanford classics professor Ian Morris says:
Mycenaean material culture dominated about 100,000 square kilometers, covering the modern nation-state of Greece (except its northern part) with enclaves on the west coast of Turkey. The population of this area was perhaps a million.2
This uncertainty is because our knowledge of the period is too piecemeal to support reliable, precise estimates. A significant corpus of contemporary written records do exist, but it is fragmentary and specific to regional polities. Population estimates thus necessarily rely upon extrapolations from archaeological surveys of settlement sites. Yet not all of these has been, or can be, found.
Nonetheless, relatively rigorous estimates have been created for specific regions where surviving records or archaeological attention have been comparatively more concentrated.
Perhaps the best studied case is in Messenia, where the palatial state of Pylos thrived. Between 1962 and 1968, the Queen's professor Richard Hope Simpson and Minnesota historian William Andrew MacDonald led an interdisciplinary effort to survey the region. Their pioneering effort is known as the University of Minnesota Messenia Expedition.
Based on the results of their survey, McDonald and Hope Smith cautiously estimated the Mycenaean population of Pylos to be at least 50,000 based on the 250 settlements discovered from the period.3 This figure has since been generally accepted. More recently in 2001, Todd Whitelaw, Professor of Aegean Archaeology at University College London, made the same estimate.4
In addition, Messenia's Palace of Nestor housed a large trove of Linear B tablets. These reported administrative inventories from the Pylian polity's final year, attesting to some 4,000 people. As a sidenote, the distinguished British Linear B scholar, John Chadwick estimated the Mycenaean Messenian population might have been about 100,000,5 though this is not supported by existing archaeological evidence.
By far the largest collection of Linear B texts are found at Knossos, on the island of Crete. Using a multifactored approach that combined the written records and archaeological surveys, Richard Firth in 1995 proposed a total of 110,000 residents on the island in the Post Palatial period (LM IIIB).6 For comparison, the Sheffield archaeologist Keith Branigan estimates that Neo-Palatial Crete (MM IIIB) a few centuries earlier had a population of 140,000 to 160,000.7
1. Ober, Josiah. The Rise and Fall of Classical Greece. Princeton University Press, 2015.
2. Morris, Ian. "The collapse and regeneration of complex society in Greece, 1500-500 BC." After Collapse: the Regeneration of Complex Societies. Schwartz, Glenn M., and John J. Nichols, eds. University of Arizona press, 2010.
3. McDonald, W. A., and Hope Simpson, R. "Archaeological exploration." McDonald and Rapp, eds. The Minnesota Messenia Expedition: Reconstructing a Bronze Age Regional Environment. University of Minnesota Press, 1972.
4. Whitelaw, Todd. "Reading between the tablets: assessing Mycenaean palatial involvement in ceramic production and consumption." Sofia Voutsaki and John T. Killen (eds.), Economy and Politics in the Mycenaean Palace State. Cambridge Philological Society, 2001.
5. Chadwick, John. "The Mycenaean Documents."McDonald and Rapp, eds. The Minnesota Messenia Expedition: Reconstructing a Bronze Age Regional Environment. University of Minnesota Press, 1972.
6. Firth, R. "Estimating the population of Crete during LM IIIA/B." Minos: Revista de Filología Egea 29 (1994): 33-56.
7. Branigan, Keith. "Aspects of Minoan urbanism." Urbanism in the Aegean Bronze Age (2001): 38-50.