In the USA, anyone can buy property. Your citizenship is not an issue or hindrance to buying. The US government does review the source of money entering the US.
All you need is to select a qualified Real Estate Agent to be your Buyer's Agent. I suggest you chose one that has worked for foreign nationals before.
Your US travel visa (or green card-work permit) will tell you how long you may visit the US and thus limits the amount of time you can spend in the US each year. http://travel.state.gov/visa/
Use this website to determine how long you can visit the US each time. And contact them .
There has been much talk about the new EB-5 Visa. http://en.wikipedia.org/wiki/EB-5_visa
This program applies only to foreign nations who are making investment in job creation projects/businesses. Application can take 2 years.
But as I said, none of this prevents you from buying property.
Please contact me if you would like more information. I am a Realtor covering Central Arizona, including Phoenix, Scottsdale and the surrounding cities.