Ancient Crete had not one but three writing systems at roughly the same time: the Cretan hieroglyphics, Linear A, and Linear B. Linear A and the Cretan hieroglyphics have not been deciphered, but Linear B has.
Linear B has about 200 glyphs, about half syllables and half logograms. It was used almost exclusively for administrative records. Most of the logograms have to do with food somehow — meat animals, staples, and vessels — as would make sense for record-keeping in an era when commercial transactions would mostly revolve around food.
The first evidence of Linear B was on engraved gemstones that caught the eye of a British archeologist, Arthur J. Evans. He scoured Crete looking for more evidence of that script, and found it in a big way at the palace of Knossos in 1900. Several people tried but failed to decipher the script, but in the early 1950s, Michael Ventris and John Chadwick succeeded in deciphering the script. They were quite surprised to discover that the language was Greek — this put the earliest date of Greek to 600 years earlier than anyone had thought.
Part of how they managed to decipher Linear B was through prior work by Alice Kober. Kober recognized that there were common patterns of starting glyphs, and common patterns of ending glyphs, and surmised that the starting glyphs were roots and the ending glyphs were suffixes. There were some glyphs that would appear in the middle of words which didn’t belong to either the common roots or to the common suffixes. She correctly surmised that those were bridging syllables (which appear in some other languages: the beginning consonant of the bridge syllable belongs to the root and the ending vowel belonging to the suffix). From that, she was able to make a table of consonant identifiers on one axis and vowel identifiers on the other axis, with the appropriate syllable glyphs in each cell — despite not knowing how any of the consonants or vowels were pronounced!
Ventris and Chadwick then matched up the written syllables with some words in Greek (place names, as it turned out), and from there were able to build and build until they had the whole syllabary.