Kit, I understand your desire to have a unique trip and not just do the "touristy" things, but many of the cities you've listed don't have adventure activities due to their locations. There are places in each country that offer adventure tours, but they are certainly off the beaten track and involve some co-ordination to get to.
Switzerland has a lot of things like paragliding, bungee jumping, mountain biking, hiking, etc) and Interlaken is home to most of the companies offering these.
Herrbert's suggestion for
Austria is spot on - the whole region in the Tirol valley is teeming with white water rafting, paragliding, rock climbing, mountain biking...whatever takes your fancy.
Paris- nothing close by, but there are adventure companies that operate in the Dordogne for rafting/kayaking on the river; trekking and climbing companies operate in the region around Chamonix and in the mountains of Provence (behind Nice) there is hiking and climbing in Summer and skiing in Winter; it might not be your scene, but there are bike rides available through the Loire Valley that cruise past the vineyards and castles.
Brussels- unless you call dodging Eurocrats walking down the street a sport, there's not a lot of adventure sports to be found here. Bike rides are about the only thing fitting your criteria - most of the region around here is very flat so nothing too strenuous. Here's something quirky (and cheesey!) in Brussels is Mini-Europe. It's near the Atomium (also quirky - gotta love Belgium!)just on the edge of the city and easily reached by train.
Munich- as the Tirol is only a short train ride away, this is the best area in the region for adventure sports so see Austria for details. Something different in/near Munich is the Andechs Monastery (
Kloster Andechs). Monks still brew the traditional strong beers produced here that were originally developed as "food" for Lent and you can take a small tour of the gardens and monastery too. Look at
www.andechs.de/ for detail, there is a train and shuttle bus to get you there and it's worth the trip.
Amsterdam- again, it's pretty flat around here so no mountain biking or white water rafting (unless the canal boats create a big wake!). There are bike hire companies in the city where you can hire an Omafies (grandma bike!) and cycle around the city, just watch the trams!
Vienna- it's all about the culture here. The best I can suggest is a trip up one of the towers of Stephansdom (St Stephan's Cathedral), or a tour of the crypt underneath. Or perhaps visit the Kaisergruft (Kapuzinerhof or Imperial Burial Vault) on Tegettstrasse to pay respects to the Hapsburgs. If you want to have some fun, head out to Prater - the amusement park with great rides and food halls. Locals and visitors all go here because it's good fun.
Prague- I believe there are skydiving companies that operate out of Prague, but I'm not 100% sure of that. Otherwise, there is white water rafting on the River Vltava by Csesky Krumlov. I haven't been rafting in the Czech Rep so don't know what grade the river is (guess Summer would be around a 3 as snow would have melted by then and the river would be calmer), but it would be worth checking what companies are around. I think there is also rock climbing just outside Prague (Srbsko), but check on that.
Budapest- The only thing I'm aware of is the Matyarshegy Cave near Budapest where you can take guided tours through the caves which are pretty spectacular (I'm told, I'm yet to go). I'm guessing there will also be Rock Climbing, might be best to check directly with Hungarian Tourism Association
Here's a link for mountain biking in Europe that might help:
mtb.live555.com/mtb-europe.html. With any of the companies, make sure they have saftey and emergency procedures in place and that you are provided with all of the safety equipment you should be - you shouldn't have to ask for a helmet, life jacket, etc, they should automatically be provided or you bring your own. And don't be afraid to ask questions about their procedures.
Don't discount the museums and other "touristy" things - think about what visitors to your region like to see and do and they may prove to be considered "touristy" too, but really are "must-see" things. You might not be able to jump off a mountain in Amsterdam (distinct lack of mountains!), but you also can't visit the van Gogh musuem in another city either. So try to balance your trip with a bit of both, then it really will be unique! ;D
