Chances for non-U.S. founders to succeed in Silicon Valley

If you're based in U.S., it is not very hard to get in touch with the right people in order to transform an idea into a product. It's easy to get to conferences, to apply to different startup programs etc. But what about entrepreneurs-wannabes from other countries (an European country, for example)?

I'm sure there are a lot of smart people around the world, capable of having great product ideas, who at least once asked themselves: "How would it be to get to Silicon Valley and make this thing happen? ". But how hard is to make that possible? What are the chances for some non-U.S. founders to make themselves heard and get to Silicon Valley, the place where a startup idea is always welcomed and has the most chances to obtain funding and fully benefit of networking?

Supposing you have a potentially-awesome idea, which are the biggest obstacles to getting to Silicon Valley? Are there any relevant examples of people who made it in a situation like this?

It is difficult as sometimes the USA gives the impression that overseas entrepreneurs are just like people who sneak across the US-Mexico border in the middle of the night. Fortunately, this is changing, with the Startup Visa initiative, etc.

I'm in the UK and would love to start my next technology business in Silicon Valley, but it is very difficult to get a suitable visa (I want to do it legally, not by any dubious means).

Alternatively, it is possible that you have a good start-up ecosystem near you.