Summary: This article presents the results of my research carried out from 1977 on the Cretan hieroglyphic script. One of the most famous hieroglyphic inscriptions, the so-called Phaistos Disk (1700 B.C.), is written in an Old Greek dialect. The principle of acrophony used in this script and the fact that the Cretan linear A and B scripts are pure simplifications of the Phaistos Disk script indicate, that the Greeks must have played an important part in the Minoan civilization. The Greeks must have arrived in Crete long before the assumed date of the beginning of the Mycenaean period in Crete (1450 B.C.).