By comparing the translations, one can note to see the "Great Cucumber Divide", a line running from the North Sea to the Adriatic and dividing the continent in half. Almost everywhere in the Eastern, Northern, and Central Europe, the word for "cucumber" is a derivative of the Greek αγγούρια ("unripe"; all examples here and below are in plural, and in an oblique case); cf. German Gurken, Swedish gurka (similarly in other Germanic languages, except for English), Czech okurky, Slovak uhorky (similarly elsewhere in West and East Slavic), Latvian gurķiem, Finnish kurkkujen, etc.
Admittedly there is a strange non-αγγούρια island in the Balkans, with the Bulgarian краставици and obviously related Romanian castraveți. (Outside of the EU directive, we also find the same word in Albanian (kastravecë), Maceodnian, and Serbian/Croatian). So in this case the Balkan Sprachbund has a common word, but it is not the same Greek word that's common throughout half the Europe!
The south-western half of Europe is much less homogeneous. Spanish and Portuguese have pepinos, which comes, ultimately, from the Greek πέπων "melon" (as does the English pumpkin).
The English cucumber, via the French concombre is said to be derived from the Latin cucumis. In Italian, this word survived too, as cocomero, but there it is more likely to mean "watermelon" than "cucumber"; the apparently more standard Italisn word for "cucumber" preferred by the EU bureaucrats id cetriolo, which also happens to be a Greek loanword - with the original Greek meaning "citron"!