Personally, I think the American designer who had the biggest impact on European golf is Robert Trent Jones, by a considerable distance.
Jones and his firm built a number of high profile courses across Europe at a time when golf was really starting to grow in those countries, both as a tourist proposition and among local players. But his influence extends far beyond his firm's own courses.
Some of the key players in today's European design sector were Jones trained in one way or another: Cabell Robinson in Spain to David Krause in Germany and Jeremy Slessor, the boss of European Golf Design, the European Tour/IMG joint venture that is by a distance the continent's largest design shop.
More than that, though, the Jones style of design - called it the modern movement if you like - remains the dominant form across much of Europe. We see courses that have large greens, lots of water and big, complex shaped bunkers all over the continent, some created by American firms, but also lots by Europeans who came to the game at a time when Jones's ideas were dominant.
Look at the European course rankings. Obviously Valderrama is a Jones/Rulewich course, but places like PGA Catalunya, though created by Europeans, reek of his influence.
Even today we see new courses being built in this image - I would say that the International course in Amsterdam, which a number of GCAers saw this summer, is clearly a course in that mould.
We see little of a direct Dye influence in European golf, except on a very facile level - the island greens that are everywhere, the Whistling Straits fantasy of Sand GC in Sweden.