You need to make sure any pets are vaccinated, including rabies, although I don't think there is much rabies here in Greece.
Speak to a local vet when you arrive and find out what precautions you should take against local diseases, mostly borne by insects. Pets may need tick collars and regularly treatment to help them being bitten by sand-flies, for instance.
We brought two dogs with us when we moved and they have adapted well and I know people who have arrived with cats as well.
Just be aware that the Greek attitude to domestic animals is not the same as you are used to. Many Greeks are afraid of dogs and believe that they should be tied up outside all the time.
It is one of the things I don't think I will ever get used to here, although attitudes are changing slowly.