Short of applying for residency in a particular country, there's no tourist extension of the Schengen visa. (Trust me: I've had to rework my own plans to accommodate this.) Many people have suggested that, as an American, you can just choose to ignore those restrictions, since Americans aren't closely scrutinized at border crossings. And many have given first-hand accounts of either traveling or living in Europe well past the three months and not running into any problems at all. Just bear in mind that such a route is illegal, and not entirely fail-safe, and if you get caught overstaying the three-month period, you'll have to deal with the fall-out.
My suggestion: As long as your plans aren't set in stone (I can't imagine you've booked a year's worth of flights and hotels), shuffle up your itinerary so that your time is split between Schengen and non-Schengen countries in three-month chunks. As a point of reference, the 15 Schengen countries are: Austria, Belgium, Denmark, Finland, France, Germany, Iceland, Italy, Greece, Luxembourg, Netherlands, Norway, Portugal, Spain and Sweden. You'll notice that the UK isn't included there; neither is most of Eastern Europe. (Neither is, oddly, Switzerland.) For that matter, neither is North Africa - not a bad stop during the winter. I'm side-stepping Schengen by spending my first three months in Spain and Portugal (admittedly, if I was hell-bent on it, I would've included more of Western Europe); then ducking down to Morocco, Tunisia, Egypt, Israel, and Turkey for the winter; then coming back through Eastern Europe; then back to Western Europe. Entirely feasible - and, on second glance, more sensible than my original plan.
On paper, you might think your plans have been ruined. But once you start to think through the many attractive options, you'll realize that - even if it's not in the order you first planned - you can still see all of your sights in a year.