Here's a fairly astonishing survey from the Pew Global Attitudes folks. When they polled various countries on their general favorability toward the U.S., attitudes were improved since Barack Obama's election, but for the most part not improved dramatically. European countries were generally far more favorable toward the U.S. compared to last year, but most of the rest of the world was only moderately more favorable.
But they also asked if respondents were confident that the U.S. would "do the right thing," and the results there were stunning. With the sole exception of Israel, every single country registered an increase in confidence toward the U.S. A few of the increases were moderate (Pakistan, Lebanon), but most of them were stratospheric (Egypt, Spain, Canada, Japan, Brazil).
Time will certainly erode this goodwill. That's just the nature of these things. But for now, the rest of the world has a spectacularly improved view of how they expect the United States to act on the world stage. As Dan Drezner says, this is a hard measure of Obama's — and America's — newfound soft power.