I purchased a new car and have registered it in Germany. I want to use it to travel to the Czech Republic, but I haven't received the Green Card from my car insurance company.
Would this be a problem?
If you believe the UK MotorInsurers'Bureau page on the green card the green card is not necessary within the EU if the car is registered and insured in the EU:
I have been told that a Green Card is not necessary for travel in Europe. Is this true? It is correct that a Green Card is not required by law to cross borders within the European Union. This is because every EU country complies with the First Directive on Motor Insurance which says that every insurance policy issued in the EU must provide the minimum insurance cover required by law in any other EU country.
Green Cards are also not necessary for some non-EU countries who are signatory to Section III of the Internal Regulations, which is an international agreement between Green Card Bureaux. These countries are Iceland, Norway, Switzerland (incorporating Liechtenstein), Andorra and Serbia. Although a Green Card is not necessary for these non-EU countries our advice is to check the insurance position with your insurers before you travel. Our understanding is that a UK insurer is not legally obliged to provide cover for non-EU countries, although some may choose to do so.
Similarly AngloInfo Berlin states it is not necessary to have a green card, but recommendable to carry some proof of insurance:
It is not essential for an EU citizen to have a Green Card when travelling within the EU and certain other European countries, though if you do not take one you should instead carry your Certificate of Insurance. However, a Green Card can serve as easily recognisable proof of third party insurance, for example in the case of an accident, when travelling abroad.
Finally the UK governments nidirect page lists countried you need no green card for and such where it is needed:
A Green Card is not required by law to cross borders within the European Union and some other countries. This is because all EU countries and certain other countries comply with the first directive on motor insurance, which says that every insurance policy issued in the EU must provide the minimum insurance cover required by law in any other EU country.
The countries that don't need a Green Card are: Andorra, Austria, Belgium, Bulgaria, Croatia, Cyprus, Czech Republic, Denmark, Estonia, Finland, France, Germany, Greece, Hungary, Iceland, Ireland, Italy, Latvia, Lithuania, Luxembourg, Malta, Netherlands, Norway, Poland, Portugal, Romania, Slovak Republic, Slovenia, Spain, Sweden and Switzerland.
The countries that do need a Green Card are: Albania, Belarus, Bosnia and Herzegovina, Former Yugoslav Republic of Macedonia (FYROM), Islamic Republic of Iran, Israel, Moldava, Morocco, Russia, Serbia and Montenegro, Tunisia, Turkey and Ukraine.
Note that the accepted answer is not universally true. To quote Wikipedia,
Within the European union, some countries (such as France and Belgium) have kept the Green card as their national/domestic system of insurance, which make the green card a compulsory requirement in those nations.
In France, failure to present the green card during a traffic check can result in a 150€ fine (source in French).