"We can't just have a few senators spoiling everything for everyone here," Senate Majority Leader Harry Reid, D-Nev., said after the vote.
Lee's unsuccessful effort to derail the bill isn't likely to end anytime soon, though.
The senator says he may demand a 60-vote threshold for some amendments, though he wouldn't preview his plans or say whether he, too, would try to attempt to change the bill.
Despite losing the vote Thursday, Lee said his battle was worthwhile since it ensured that nothing would pass unless it had "broad-based, bipartisan consensus."
"We're pleased, nonetheless, that we were able to get more debate, more discussion, and more input from the American people," Lee said in an interview.
The Utah Republican disputed remarks including those from Sen. John McCain, R-Ariz. that the tea party trio shouldn't try to filibuster the legislation.
"At most that is referring to a matter of partisan political strategy. This is not about partisan politics," Lee said. "This is a policy based argument."
The group's effort, though, was met with criticism, even from the right-flank concerned that the protest shifted the debate from the merits of gun control to Republicans appearing to be the party of no.
The conservative Wall Street Journal editorial board took after Lee and his tea party buddies, calling their tactic a "misfire."
"If conservatives want to prove their gun-control bona fides, the way to do it is to debate the merits and vote on the floor," the Journal said in an editorial Thursday. "They can always filibuster the final bill if they want to, but it makes no sense to paint themselves into a political box canyon before even knowing what they're voting on."
On the other hand, Lee's public fight waged on cable TV, on the Senate floor and through social media may appeal to Utah's many gun-loving voters.
"He hurt his party but helped [himself] with his constituency," says James Thurber, director of the Center for Congressional & Presidential Studies at American University in Washington.
Sarah Binder, a senior fellow at the Brookings Institution who studies Congress, said the Lee effort may not make much of an impact with Americans since in the end it failed.
"The story line today is 'Senate moves on to gun debate,' " Binder said, noting that the one place people might take notice is in deep red states like Utah.
Hatch said he voted against moving forward because the issue "deserves a thoughtful debate."
"But the fact is that we don't even know what's in the gun-control legislation because the text of it hasn't been made public," Hatch said.
Actually, the text of the bill the Senate was voting to take on is available, though language for a compromise amendment likely the centerpiece of the eventual legislation isn't out yet. The compromise crafted by Sens. Joe Manchin of West Virginia and Pat Toomey of Pennsylvania, would expand background checks only for commercial gun sales, including those advertised online and at gun shows, but not for private weapon purchases.
For his part, Lee says he plans to make sure the bill and whatever amendments are offered don't just zip through the Senate.
"People throughout Utah are understandably and justifiably concerned about legislation that threatens to restrict the rights of law-abiding American citizens while doing little or nothing to deter violent crime," Lee said.
Lee's protest has earned him some plaudits and plenty of attention. He's scheduled to be on NBC's Meet the Press on Sunday.
email@example.com Obama congratulates Newtown families on gun vote
President Barack Obama is lauding the role the families of the Newtown shooting victims played in pressing the Senate to move forward on gun control legislation.
Shortly after the Senate voted Thursday to proceed on the White House-backed measures, Obama spoke by phone with the families, many of whom have been in Washington lobbying congressional lawmakers.
"The president congratulated the families on this important step forward, noting that the bipartisan progress would not have been possible without their efforts," White House spokesman Jay Carney said.
Carney said Obama told the families he would "keep fighting for the votes they deserve."
The Senate rejected a conservative effort to block debate on the gun control legislation. The 68-31 vote lends momentum to Obama's efforts to crack down on gun violence following the December school shooting in Connecticut. Twenty children and six adults were killed in the shooting.
Carney called the vote "very important," but said it was just the "first stage" in the effort to pass anti-gun violence legislation.
The Associated Press