Its a tricky system. You can't get a credit card without having a credit history. You can't get a credit history without getting a credit card.
Also, its nice to have a credit card for expenses when you first arrive; however, you may not immediately have a Social Security Number (SSN) which might cause problems with most banks.
Keep in mind that a lot of foreign banks issue US dollar denominated credit cards, but most of the time this will not help you because it is still issued by the foreign bank. Your goal here should be to get a credit card and start building US credit history, which can help you qualify for loans, mortgages, cell phones, etc.
Here are my suggestions:
HSBC - For example, if you are from the UK / Hong Kong and bank with HSBC, you can probably get help with a US credit card - especially if you qualify for the HSBC Premier service (to qualify you need to have $100,000 or more in deposit/investment assets.
RBC Bank - Given that I am from Canada, I was able to easily make the transfer over (even before I moved to the USA). RBC Bank (the US bank subsidiary of RBC Royal Bank in Canada) was able to grant me a credit card based on my Canadian credit history. I have Canadian colleagues working elsewhere in the United States and they did the same thing with TD Bank (the US bank subsidiary of TD Canada Trust).
For those that are interested, I chose the RBC Bank Visa Signature Black card (link). Comes with no annual fee. I obtained the card about 6 months before I moved to the US and did not need a US SSN or US address. As an aside, if you are not intending to live in the US and simply visit a lot (from Canada), this might be a great card for you. RBC Bank essentially exists to service Canadians living/playing/visiting in the US, so I suggest checking it out.
American Express (best option) - American Express offers a very helpful global transfer program if you have an American Express card in Australia, Canada, Hong Kong, Singapore, New Zealand, and the United Kingdom I had an Amex card in Canada. Basically, Amex required a home address and a US phone number as proof; I believe they even wanted to call my US bank to verify that I was living in the US (i.e. my RBC Bank account in the US had a US address on file). It was a pretty seamless process (link). My Amex was sitting in my mailbox the day I moved into my apartment in the US.
Other options to get your first credit card
Unsecured credit cards with Federal Credit Unions - I contacted a credit union based in Dallas, Texas called Advancial. Theoretically, you need to be an employee of certain organizations to become a member; I mentioned that a friend of mine worked at one of the companies on the list and I was able to become a member without any issue. However, I needed a SSN - but surprisingly they did not pull my credit history (given that I effectively had no credit history anyways). They simply wanted a copy of my passport, work permit, and my signed job offer. My guess is other credit unions would probably be just as helpful.
Secured credit cards with banks and credit unions - Your last resort should be to try to get a secured credit card. This is where you give a bank say $500 dollars and they give you a credit card with a credit limit of $500 dollars. As you spend on the card, and pay off the bill on time, you eventually build your credit score as you have a history of being a good borrower. I ended up not needing to go this route, but I had some colleagues at my office from eastern Europe who had to go this route.