WASHINGTON (CNN) -- The House of Representatives voted 223-201 Thursday to require most U.S. troops to leave Iraq by April 1, 2008.

House Speaker Nancy Pelosi, shown with President Bush in March, says the American people are demanding "a new direction."

President Bush vetoed a war-spending bill with a similar withdrawal date in May and has threatened to spike any new effort to set a timetable for a U.S. pullout. His Republican allies in the House said the new measure has no chance of passage.

But House Speaker Nancy Pelosi, a California Democrat, said Thursday's mixed report on the progress of the war shows it's time for American troops to come home.

"President Bush continues to urge patience, but what is needed -- and what the American people are demanding -- is a new direction," she said.

Earlier Thursday, Bush said a report on U.S.-set benchmarks for Iraq shows "satisfactory progress" in eight areas. He admitted that there is "more work to be done." Watch benchmark report, House vote factor into Iraq debate »

During his news conference, the president commented on the nation's psyche, declaring, "There's war fatigue in America. It's affecting our psychology. I understand that.

"This is an ugly war. It's a war in which an enemy will kill innocent men, women and children in order to achieve a political objective. It doesn't surprise me that there is deep concern amongst our people."

The president said it is not Congress' job to make decisions regarding the war.

Before the House vote, Rep. John Murtha, a Pennsylvania Democrat, told CNN's Wolf Blitzer: We're the people who decide when to go to war, whether the war should be funded.

"Now, when [the Bush administration] keeps making mistakes as they have made, we have to intercede. The public spoke in the last election and said clearly we want the troops redeployed."

"We are wasting the time and trying the patience of the American people for no useful purpose," said Oklahoma Rep. Tom Cole, one of four Republicans who voted for the measure.

Rep. Jay Inslee, a Washington Democrat, said the United States has given Iraqi leaders "a reasonable chance" to work out their differences, and it was time for American troops to come home.

"The moral obligation to Iraq has been completed," he said. "The moral obligation to our families now needs to be honored."

Four Republicans joined 219 Democrats to pass the bill, two more than backed a similar measure in March. But 10 Democrats broke ranks to oppose it.

Minority Leader John Boehner, R-Ohio, said most Republicans are unwilling to challenge Bush before a September report from Gen. David Petraeus and Ambassador Ryan Crocker, the top U.S. officials in Iraq. In a closed-door meeting Wednesday, he called Republicans who break with the president "wimps."

"It was a way of illustrating the point that we ought to give the generals on the ground and our troops a chance to succeed," Boehner said.

But Rep. Jim Clyburn, the House Democratic whip, said the vote shows Republicans need to resort to "intimidation" to keep their caucus in line.

"The name-calling that other side has resorted to, I think, is beneath the dignity of the men and women who find themselves in harm's way," said Clyburn, of South Carolina.

The vote came the same day that the White House delivered a mixed report on the progress of the 4-year-old war, concluding that the political progress of the Iraqi government is lagging behind military gains. Across the Capitol, the report added new fuel to a similar debate in the Senate, where a leading Republican senator pronounced himself "disappointed" in the results.

"That government is simply not providing leadership worthy of the considerable sacrifice of our forces, and this has to change immediately," said Sen. John Warner, the influential former chairman of the Armed Services Committee.

A broader report by the top U.S. officials in Iraq -- Petraeus, the top American commander, and Crocker -- is scheduled for September, and Bush told reporters he would consult with Congress about "the way forward" at that point.

Arizona Sen. Jon Kyl, a Republican, told CNN, "I think any judgments at this time, one way or the other, are quite premature."

But Senate Majority Leader Harry Reid of Nevada said, "The time to do this is now, not September."

"We're told, 'Good progress is being made. Wait till September. Good progress is being made.' How many times over the last 4½ years have we heard this?" Reid asked.

The Senate's Democratic leaders are using a Defense Department authorization bill as a vehicle to consider several amendments designed to force Bush to change course in the war.

Republicans have so far managed to use procedural roadblocks to head off those measures -- but faced with a U.S. death toll of more than 3,600 and deep public opposition, several GOP senators have wavered in recent weeks.

Three Republicans -- Chuck Hagel of Nebraska, Olympia Snowe of Maine and Gordon Smith of Oregon -- have co-sponsored a Democratic amendment that parallels the House bill.

But Democrats are likely to need at least eight more Republican votes before their proposal comes to a vote.

Warner and six other Republicans voted Tuesday for an amendment that would have required U.S. troops to spend a month at home for every month deployed. Two more -- Sens. Richard Lugar and Pete Domenici -- have called on Bush to change course, but have opposed Democratic calls for an American withdrawal. E-mail to a friend