What are some signs of Venetian and Genovese influence in the Greek islands (and Cyprus)?

Not aware of any Russian influence; 7 years was short, and I’m not aware that the Russian presence was substantial. The British did leave behind ginger beer and cricket.

Venetian and Genoese influence that I know of includes:

• Substantial Italian, Venetian and Genoese lexicon in the dialects of the Greek islands. When my grandmother told us off for being noisy, she’d tell us not to make ντραβάγια [dravaʝa]. That word is just travaglia: travail.
• On Crete, Venetian-era fortifications. The huge land walls around Iraklion had to withstand a twenty-year siege. In my home town of Sitia, the Venetian fortress (Kazarma: Cas’ Arma) is still the most prominent landmark. It helped that the town was depopulated for two centuries afterwards.
• The  literature of the Renaissance was substantially based on Italian models. The only enduring legacy of that was Erotokritos, which was popularised as a chapbook, and is still sung on Crete (there are accounts of people who had memorised all 10,000 verses). It also became a distant ideal for Modern Greek literature, of what could have been if the pedants had not gotten their way.
• In the Ionian islands: Western style folk music, including barbershop-style harmonies, and mandolins. Unsurprisingly, when I was a kid and Greek TV featured folk music from throughout Greece, Ionian Islander music was not included. Did not fit the stereotype.
• Greek Catholicism, although that’s limited to only a couple of islands (Syros being the main instance).