It cannot be made a requirement for an airline to demand this information BEFORE a flight from the UK to Greece. This is clearly laid out in the EU's information service summary of the Directive:
http://europa.eu/rapid/pressReleasesAction.do?reference=MEMO/10/349&format=HTML&aged=0&language=EN&guiLanguage=enThe key words here are:
"Such personal data are typically taken from the machine-readable part of passengers’ passports and forwarded to the authorities after the completion of check-in."
Note the "after".
It's important in this age of identity theft that we don't needlessly spray information such as passport numbers across possibly compromised internet links, and I would urge anybody feeling that they want to be bullied into doing this to do it by telephone. This is basically a lazy catch-all for airlines in case people are flying to (say) the USA, where there are additional requirements. If you are an EU citizen you have a right to travel with adequate documentation at the border.
Looking at our Thomson's booking confirmation, this request is at the very bottom, after in-flight meals, seat reservations, and other trivia. If it's a show-stopper, why isn't it in the initial booking phase?
The answer to that is that it's easier for them to request it, even if it's not necessary. I doubt if this information is actually used for European flights until after check-in. After all, I can book with whatever details I like. Only at the desk is this information validated. Do you really think that Thomson's and EasyJet are transmitting this stuff to Kerkira airport the second you submit it?