The Republic of Cyprus is now a member of the Pet Travel Scheme which allows pet carnivores (dogs, cats and ferrets) from any of the countries covered by the scheme to enter Cyprus without quarantine provided they meet specified anti-rabies, blood sampling and anti-worm/anti-tick hygiene requirements (note: UK pets can enter Cyprus on the Pet Travel Scheme without the need for a rabies blood test. However, they will not be allowed back into the UK without quarantine.)

Northern Cyprus is not a member of the EU and is not covered by the Pet Travel Scheme. In order to bring pets into Northern Cyprus you need to obtain an import permit from the North Cyprus Veterinary Service. Cats and dogs must have a health certificate signed by an official government officer in their home country, and must be quarantined for six months on arrival in Northern Cyprus, after being vaccinated against rabies and treated for echinoccus and ectoparasites. Certain breeds of dog cannot be brought into Northern Cyprus.

Added March 2007 by Mrs J Hall:
Your outline advice on taking pets to Cyprus seems clear and straightforward, BUT there is no mention of possible hazards for animals once they are there.
My stepdaughter recently took her two cats to Cyprus, where they were poisoned within a couple of weeks. If it's general practice for anyone to put down poison (to keep strays down?) I would like to make other people aware.