The only background information I do have is from talking to an immigration attorney myself for when I tried to hire someone from NZ two years ago. Things may have changed since. That said, each case may also differ greatly.
As far as I'm aware, you can try to apply for a HB-1 VISA which will allow you to work in the US if a company sponsors you for that VISA and the requirement is you must have a degree that applies to the position in which you were hired for until you can apply for and receive your Green Card status (separately), which can take up to 5 years. I didn't hear anything about a PhD requirement. I believe a bachelors is sufficient but you might want to double check on that.
Your toughest challenge will be finding a job relating to your degree and a company willing to sponsor you for the whole nine yards.