Since 1st May 2004, the citizens of Czech Republic can enter and stay in territories of European Union Member Staes on base of valid travelling document or identity card only. Since 1st January 2006, there is valid new Law regarding travel documents (No. 559/2004). According to this, you can use, as valid travelling document for journeys into European Union Member States, also personal identity cards, with all data readable by machine way.
According to the statement of Interior Ministry of Czech Republic, the identity card is a travelling document for the holder only. So, the Ministry does not recommend to persons, younger than 15 years of age, who is travelling together with their parents and is written in their identity cards, to travel without his own travelling document, on base of this registration in identity card only.
Since 1st May 2004, the drivers can use, as a document, valid driving licence, whereas the international driving licence will not be required by Greek officials. Nevertheless, the technical certificate, as well as so called green card, as a document regarding vehicle insurance, are required. The animals have to be equipped with international vaccination certificate.
Information and warnings on travelling abroad is to be found HERE.