In a somewhat different vein that doesn't seem to be on your radar, I'd highly, highly suggest Switzerland. We spent a week in Romandie, the French-speaking part of the country. It's a fantastic place, and there's absolutely no need for a car. The trains are all reasonably priced and ALWAYS run on time. I'm not exaggerating. Once the train left two minutes late, and the engineer apologized profusely and (here's a shocker) we arrived on time. We flew into Geneva (nonstop from Washington Dulles, IAD) but based ourselves in Lausanne, and our hotel was right across from the train station. Although I know this usually has a bad stigma, it was a nice part of town, and we could leave our room ten minutes before our train was scheduled to leave and make it easily. Connections from Lausanne are fantastic, because it's just after Lausanne that the main line from Geneva branches off to go to other places. Also, public transportation in Lausanne (metro and bus) is free to people staying in hotels in the city. You just walk up to the metro and hop on...it's really easy.
Also, the countryside is really stunning. We went up to the Alps twice, visited Geneva for a day, took the ferry across to Evian-les-bains, etc. In terms of going up to the Alps, you take the mainline train to another slightly smaller station, and from there take a narrow gauge train (see below) to the Alps. We also took one of the little trains to Gruyeres to see cheese and the Cailler chocolate factory, both in the same day. The cheese factory wasn't necessary, but it's right near the Cailler factory, which was absolutely worth it. I think I may have some pictures on this computer, but I'm not sure. I might take a look... Oh, and we only went to one museum our whole time there
. It was an art museum, which I'm not usually that into, but I have to say that it was a great museum. It was in Lausanne.
Actually one last story. We were on the train back from the Alps when we decided "meh, why not get off at some random stop and take the next train down? It's only an hour until the next one so if it's terrible, oh well." We jumped off and explored the lovely Swiss mountain town of Troistorrents. We wandered around for a while and found a store, bought some local bread and cheese, which was, needless to say, fantastic, and went to sit down and much on it for a bit. This was a Friday, and it turns out school was just about to get out, so a few minutes later, a million kids raced out of what was apparently into the school and into the shop we were just in to buy their Friday candies. This was probably my favorite experience of all in Switzerland, just because I felt like I was seeing the real Switzerland, and not all the museums, which you mentioned you're not as much a fan of. Point being, you might be interested in Switzerland. It's fantastic for the sort of "off the beaten track" sightseeing, because the public transportation is amazing and relatively inexpensive, and it's really easy.
Sorry for the rambling...feel free to ask questions here / pm if you like.
And here's the picture of the train as promised...at the biggest station on the route aside from the terminus where you make the connection between the mainline trains and these. I didn't take this picture, but isn't the train cute?
Often they hook two of them together, but they really are little mountain trains!
Trust me on that... 
