Bringing a pet into Luxembourg is actually not difficult, so long as your vet in your residing country understands the requirements. We recently moved our 9 year old Golden Retriever from the U.S. to Luxembourg and the process was relatively smooth. I say this because we did have a relocation company helping us a bit along the way. The main requirement is that your dog/cat must be micro chipped. My understanding is that it needs to be an International microchip so that your pet can be accurately identified no matter what country he/she is in. Regardless of whether your pet has been vaccinated for rabies recently, your pet will need to be re-vaccinated for rabies after having the microchip implanted. At this point, your pet cannot travel for at least 4 weeks. Therefore, plan ahead on when your travel/entry into the country will be since your pet will not be able to travel until after that point. Once this period is over, your pet will need to go back to the vet to have a health certificate completed. (There is also an EU form that needs to be completed by your vet at this time. Many vets do not do these forms, so call ahead to make sure your vet is qualified. Our vet wasn't, so I had to locate one who was.) Your vet can start the paperwork process when you have the chip implanted and the vaccination administered, but he/she will need to see the pet after the 4 week period in order to complete the paperwork. If you are coming from the U.S., you will also need to have a USDA certification or stamp for your pet. Your vet should be familiar with this paperwork and submit it on your behalf prior to your travel date. This paper is required by the pet transport company. Once the form is completed, you must travel within 10 days, or the whole paperwork process will have to be repeated, including the health examination. I was told, since I was moving from Texas, that this process can take some time depending on what state you are coming from. Our vet was not quite up to speed on all of the details, so our relocation company took over at this point and obtained our USDA paperwork through New York.
Unlike the UK, your pet does not need to be quarantined. Only countries requiring a Rabies Titer (I think that is what it is called) will require your pet to be quarantined. There are several websites that explain that process, but Luxembourg and the EU in general does not require it.
Once you are in Luxembourg, you need to have your pet registered with your Commune. Schedule an appointment with a local vet here and they can help you with the paperwork. Also, if you plan to travel within the EU with your pet, you will also need to obtain an EU passport for the pet. This can also be done with your visit to the local vet.
Hope that helps!
C