Any employer will want to see your online portfolio before granting you an interview. You may be able to make some progress with companies via phone interviews, but they'll most likely want to conduct an in-person interview as well before making a job offer, so you may want to try sending your portfolio to multiple companies and line up a few interviews for when you make your way over here.

If you're coming from another country ? You'll pretty much have to walk on water. You have to be a stellar candidate in order for a company to prove that your job is something that american citizens can't do or else you can't.
But if there's a will, there's a way so good luck my friend.

Let's get real here kids...

Step 1 - Create a bitchin' online portfolio that is built to impress
Step 2 - Send cover letters and the URL of said portfolio to agencies (craigslist is a good place to find agencies looking for good designer/developers
Step 3 - If an agency is interested in employing you, and your portfolio is good enough, you can work remotely as a contractor - no need to ever set foot in the US

No walking on water necessary

And Europeans are not exempted

You do understand that many "employers" are willing to pay designers & developers US dollars - cash money, under the table right? Employers can save money by offshoring; it's very common. Foreigners can work remotely, they don't even have to set foot on US soil. Many times, foreign designers will never actually speak to their US employers - all transactions are handled via email. All of the legalities concerning visas are moot.

If you're looking for a web job. Then you need to come in as a "graphic designer" and they require a graphic designer diploma plus 3 years "graphic design experience".. it should show on your CV.

Do you work for US Immigration & Naturalization? Where are you getting this information? Who "requires" a graphic design diploma and 3 years of experience? As an American who is in charge of seeking talented designers & developers to work as contractors for various projects, I can tell you that you're really making this far more complicated than it has to be.

How much Guru-level-PHP/MySQL-developer get $ in US?

Try it and you'll see how hard it is. Unless you find yourself an american citizen to marry.
The rules have changed since 9/11

Dimmon:
I agree with ajustafresh... It's not so complicated.
Even if european citizen needs an H-1B visa to work in the US, it's NOT hard to obtain it.

atomiccafe - I find it very interesting that you keep coming back to this topic - certainly a sticky subject for you.

Evidently, you don't follow the news very closely here in America; I'm going to give you some reference points to illuminate you and the rest of our foreign friends. Here are the facts:

Have you heard of "outsourcing"? It's a fairly common practice in the US and other countries. Large corporations like Dell, IBM and Cisco hire foreign employees to do work (commonly in the IT field) because the corporations can pay an employee in India much less than what they'd have to pay a comporably qualified American.

Links:
oDesk - Outsourcing Company
Malaysia attractive for outsourcing

With regard to illegal or undocumented workers in America... Did you know that there are currently at least 12 million illegal immigrants currently living and working in America? I would argue that your point with regard to "ALL successful companies" not hiring people without the proper documents is specious and based solely on your opinion. Large corporations, even state and federally funded agencies hire illegal aliens because the United States government does not enforce immigration laws. In fact, the US Congress is trying to pass legislation that will provide amnesty for the illegal residents currently living here.

Sorry if this is getting too political; might be time to retire this thread.