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Zorba
Greece is not an exotic country, so in winter and autumn the weather is far from a tropical one. Therefore, for temperatures and rainfall the good season is between April and October (inclusive).
Let's now a bit more detail:
you are compelled to leave in July or August? in this case choose July as there will be fewer people because the Greeks also go mostly in August. If you go in summer to avoid the famous places (Mykonos, Santorini ...) who will be too busy and you spoil the visit, unless you like the crowd, which is entirely possible.And remember, it can get very hot (over 40 degrees) and this can be tiring especially if you have more than 20 years, the movement within the land will be painful, so obviously at the edge the sea you can stop for a swim. If you can not avoid in August go in preference to the end of the month when the majority is back ..
you can leave whenever you want happy lucky? then the choice will be based on other criteria.
Do you like swimming in hot water? Go in September it will be the most pleasant months: Wed heated by the summer tourists gone. Early October can still bathe in late October but it will not be chilly.
the swimming area does not interest you? leave in May to benefit the spring, small flowers that grow and mild temperatures. In June it will be warmer and the days are longer but there will be a few more people. If you go in April the villages will be deserted beaches you for you alone but you might get some rain during our stay. You'll need to bring a sweater and a windbreaker because the wind is often ( always?) very strong in April especially on islands. But often those that remain open are interesting because they work with a clientele mainly Greek. In October you can also have very good weather even at the end of the month (personal experience) but you can also have nice showers (same experience).
you want to explore an aspect of Greek life? go during the holy week of Easter processions and follow the festivities. Warning remember that the timing of the Orthodox Easter is different from the Catholic Easter ( see calendar )
another aspect? go during the National Day on October 28, you can watch the procession of schools that includes a massive crowd. There is also another national holiday in March but as I recommended this month ...