It's a hard question to answer in general, but it’s safe to say that yes, there are U.S. employers willing to go through the process of sponsorship for foreign employees, even though the recession has taken grip here as well and native U.S. citizens are unemployed.
Yes, as you say the sponsorship process can be a hassle – we’ve done it here for Irish hires in the past, and the process can be quite involved, not to mention expensive between paperwork filing fees and lawyer costs.
You mention sponsorship for a green card, and as you say, you’d need an employer to initiate this process on your behalf. (There are cases where workers do not need to secure sponsorship to obtain a green card, but the applicant must have what’s known as “extraordinary ability” which definitely doesn’t apply for recent college grads like yourself.)
It can take a long time to secure a green card through employment (more than a year), but H-1B visas for foreign workers with bachelor’s degrees or above are also available and are quicker to obtain. There are 65,000 H-1B visas available to applicants around the world each year, and though that quota used to fill up very quickly when the unemployment rate was low here, nowadays the category stays available much longer.
The H-1B visa, good for three years and renewable for another three, would also need an employer to act as sponsor. The worker could apply for a green card in H-1B status if the employer agreed to be the sponsor.
Potential employees who bring a special something to a job on offer will always be appealing. So the market here definitely is not closed to foreign workers such as yourself. You’ve just got to find an employer who considers you a right fit. This might not be easy, but it’s definitely not impossible.
You should also keep in mind the annual green card lottery. The application period for the 50,000 green cards on offer each year begins on Tuesday, October 4 and ends on Saturday, November 5. Entries are lodged electronically through a State Department website, and winners will be the next spring.
Visit http://travel.state.gov/visa/immigrants/types/types_1318.html for more information.