The locals will know you are a tourist the second you open your mouth ... and the camera round your neck is a dead give-away.
Wear what you are comfortable in - unless it is a fur coat no-one willjudge you by what you are wearing
Jeans, your favorite type of shirt or blouse, and comfy shoes.
About the only clothing I can think of that gives people away as American tourists might be those safari shirt/jackets that people wear, to carry everything around instead of wearing a practical messenger-style bag, parachute pants that make noise when you walk, or jogging pants.
Go to some webcams in the cities you are visiting and look at what people are wearing to get a good idea what we mean. N.America and Europe all seem to go to the same stores, Hilfiger, H&M, Benetton, Zara, etc.
You won't manage not looking like American tourists, so just be yourself.
Comfy shoes are really important as you'll almost certainly be doing lots of walking. Lots of places still have cobbles or small blocks (or in the case of some Berlin pavement very small stones like a mosaic, so for women high/narrow heels can be treacherous at times.
.Even if we all have the same clothing chains, the stock varies with the country to some extent (they know what sells). In general people dress differently in Berlin from Munich, differently again in Poland. And so that's how tourists of different nationalities can be identified. It applies even within Europe, throughout Europe, it's not just Americans.
When abroad (from home in Spain) we can identify Spaniards too (or if really snappy dressers they are probably Italians) and in Barcelona you can take a guess, not just at Americans, although only Americans post on Barcelona forum with the same question you are asking about how not to stand out. Europeans don't seem to bother about it!
Americans bother about it largely, I think, because the spectre of the "ugly American" that reared its head in the 50s has never quite gone away, and continues to hang over people, especially thoughtful people and politically-historically aware people. They are nervous about how they are going to strike their European hosts, and how they are likely to be treated.
Don't worry too much, joekg. As long as you behave like a civilised person towards others, you'll be all right whatever you wear. Be comfortable - especially in the footgear department.Edited: 5:12 am, March 09, 2014
Afterthought re comfortable footgear: white sneakers are a dead giveaway. :0)
I think that dress is less important than attitude. A willingness to learn a few basic phrases and not to assume that everyone speaks English goes a long way. Observe the way people conduct themselves in public places, bars and restaurants (except Irish pubs!) and act accordingly and you will be welcome everywhere, however you are dressed.
For inspiration what people wear in Germany, have a look at www.otto.de for instance. People over 50 normally wouldn't wear shorts unless they are on vacation. Also, women wouldn't wear all pink, like some do in the U.S. It's something that hasn't caught on in Europe.
Also, do not bring one of these giant water bottles with you. I often see tourists lugging around 1.5 liters of water with them. Why? The tap water in Germany is perfectly safe to drink, and a small bottle is enough, which you can refill when it's empty.
If you avoid khaki shorts and white athletic shoes with tennis socks you will already look different than about half of all American tourists. Personally I think that everyone should wear what she/he feels most comfortable in but since you asked... ;-) As said before: People over the age of say 30 just don't wear shorts over here - unless they are in their backyard, in the gym or on a beach vacation.