Having done a little more research, I've discovered that this type of event is way more popular than I imagined. The name for these kinds of events is "cyclosportives", "brevet", or "radmarathon" - stick one of those words into Google and you'll get lots more info. I found some good indexes of events around Europe at: cyclosport.com (in France), cyclosport.com (outside France), sportingtours.co.uk, cyclomundo.com, veloventoux.com.
Good indexes for Swiss events are at cycling-matters.com and datasport.com (select "cycling" from the "Sport" drop-down menu).

Of those in Switzerland, here are a few highlights:

Gruyère Cycling Tour: Formerly known as "La Pascal Richard", this event starts and finishes in Bulle (canton Friburg), and goes over the moderate Alpine passes where cantons Fribourg, Bern, and Vaud meet. Distances are 85km, 125km, or 165km, with 900m, 1700m, or 2900m of climbing, respectively. Held on August 26th, 2007. This one is around my local riding area, so I'll definitely be aiming for it.

Alpenbrevet: Based around Andermatt, this is the most serious event of those that I've looked at. The biggest route is 266km with 6600m of climbing over the Susten, Grimsel, Nufenen, Lukmanier, and Oberalp passes! The less insane can opt for one of the 160km, 120km, 102km, or 32km routes with 5000m, 3500m, 3000m, or 900m of climbing, respectively. The 102km (3000m) route is for mountain bikes. It'll be held on August 11th, 2007.

Züri-Metzgete: This is held at the beginning of October each year around Zurich. It is 68km long, with 700m of climbing.

L'Alex Moos - La Cyclosportive Valaisianne. Three moderate loops to choose from of 73km, 94km, or 105km, with 100m, 1100m, or 1400m of climbing, respectively, around Sierre, Sion, and Crans-Montana. Held on August 1st, 2007.

There is one in Switzerland that is even longer than the Alpenbrevet, called the "Wysam 333", which does a 333 km loop through the Swiss Jura mountains. You can decide to do it in either one day or two, but this event is not being held this year, you'll have to wait until 2008 to take on the challenge.

There is also: La Classique Genevoise (around Geneva in June), the Gerolsteiner Classics (one in June in Appenzellerland, and one in September around the Austrian-Swiss-Liechtenstein region), and Bison Franco-Suisse (around the Vallée du Joux region in the Swiss and French Jura in August).

If anyone has any experience with any of these events, then please let us all know about it.