For me one of the most interesting parts about traveling in Europe is visiting the museums. 
Thousands of years of history are captured in sculpture, paintings and other precious artifacts. Taking good images of those things home is harder than it might appear. First and foremost many museums do not allow you to take photos at all. The new Athens Acropolis museum is like that. They want you to purchase the guide books and that is understandable. Checking your camera at the coat locker can be an uncomfortable feeling, so be prepared. Most guidebooks, such as the excellent DK travel books will tell you ahead of time. Although, I would rather have my gear with me, as you never know what you will see on the way rather than leave it in most hotels. 
Second most museums prohibit the use of flash although the Louvre in Paris does not, which I find strange. Some museums will charge you extra if you want to use the flash. So if you still want to take pictures, you have to know how to turn your flash off. This means shooting in available light, which is often dim.
The obvious choice is to bring in a tripod—whoops, most museums hate tripods worse than they hate flashes. They tend to trip up others and block the flow of traffic. One way around this is to use a monopod. No one seems to blink at them.
The challenge is holding the camera steady enough to get good pictures. For most people this means the shutter speed on the camera has to be faster than 1/60th of a second. The in camera techniques are the ones I’m going to talk about. The first is to just use a faster lens. Yep these are more expensive right? Well not necessarily. A good Nikon 50mm f1.8 is right around $100 dollars. Yes you have to get closer and use your foot zoom, but it will be fast enough in almost any lighting conditions. Of course, a really good lens will make getting these kind of shots easier.
Two simple techniques can also help improve your stability. Lean against a wall or the side of a case and brace the camera while shooting—it makes a difference. Also proper camera hand position, really adds a stop to your shooting. Keep your arms tucked in close to your body, hold your left hand under the lens to cradle it and pull it towards you. You see people waving their arms way out in front like they were afraid of the camera. It also helps to put the camera on continuous mode and shoot a lot of shots. One of them is likely to be in focus and sharp.
Be aware of the glare from the lights and the glass of the display cases. If you can get close, use a rubber lens hood and hold it against the glass. Walk around the room and see where the reflections are minimized. Use the other members of your group to block out really obnoxious lights. Often the lights are hideous fluorescent which make the colors look bad. Luckily, Lightroom processing can really help cut through glare and correct the light as in this fresco from the buried city on Santorini.
Most museums are busy places; when you compose your photos ensure you know what is in the background as well. Too much clutter and the subject will be lost.
By walking around the room I found the walls on the other side to be a much better background. Don’t be afraid to move.
Lastly, there are crowds. They will get in your way. Be considerate of others and hope that they will return the favors. 
One bonus tip, shoot the signs for the exhibits as well. Digital is free and it will help you remember what that thing was anyway when you get back and start playing with your pictures. 