When Republicans bring up health care, they tend to list it as just one example in a litany of complaints about the “misplaced priorities’’ and “overreaching’’ of Democrats. Even many Democrats are downplaying the new sweeping overhaul, once trumpeted as a signature accomplishment, because they are skittish about being portrayed as advocates of big government.

In a sign of how rapidly the ground has shifted, the new health care law, approved in March after a year of legislative struggles and caustic town hall protests, didn’t even come up last week at a marquee US Senate de bate in California between Democratic incumbent Barbara Boxer and her opponent, Carly Fiorina.

The reason, analysts say, is clear: Jobs and the economy dominate the concerns of Americans, particularly those coveted independent voters who will tip individual midterm elections. On health care, polls find, American opinion is more evenly divided and not as easy for either party to exploit for political gain.

The early conventional wisdom that health care would be the top issue did not account for the way persistent joblessness would erode the electorate’s confidence in the economy and in the Obama administration, said Robert Blendon, a professor at Harvard University School of Public Health and a specialist on voter sentiment about medical care.

“Most Americans, right or wrong, believe that the efforts by Congress and the administration to turn the economy around have not worked very well,’’ he said. “That has made it a harder sell for health care.’’

In Massachusetts, where Republican Scott Brown won a stunning upset in January’s special election to the US Senate in large part by railing against the president’s health care plan, the issue is dwarfed by economic concerns, said Jennifer Nassour, chairwoman of the Massachusetts GOP.

“We haven’t seen any candidates making health care one of their priorities,’’ she said. “Even when Senator Brown was first campaigning, his main message was that jobs are job one. That hasn’t changed at all.’’

The issue of health care has not disappeared. Some conservative groups are conducting surgical strikes in select races. Heritage Action for America, a political arm of the Heritage Foundation, a conservative think tank, has launched a 10-day television and Web campaign to pressure Democrats to sign onto a repeal of the health care law.

Targets of the campaign include Democrats Travis Childers of Mississippi, Ike Skelton of Missouri, and Mike McIntyre of North Carolina, said Daniel Holler, communications director for Heritage Action.

“What people respond to is the issue of misplaced priorities with the economy looming so huge,’’ said Steven Law, the group’s president. “When the president and the Congress had a chance to address the number one issue, they went instead on an ideological agenda.’’

The nuanced message and limited strikes reflect a basic fact about the new law: Parts of it are popular, such as a ban on insurers refusing to cover people with preexisting conditions.

“That’s the irony of this health care reform bill,’’ said John Anzalone, a Democratic pollster who worked for the Obama campaign in 2008. “If you dissect the individual parts of this bill, people are for the individual parts. They’re not for the whole.’’

Voters are equally divided on whether health care will affect their votes in November. In a poll conducted for the Kaiser Family Foundation in August, about one-third of the voters said they’d be more likely to choose a candidate who supported the law, and about a third would more likely back one opposed to it. The rest say the issue will not affect how they vote.

Democrats plan to run on jobs and economic issues this fall, but will use the popular changes in the health care law to defend themselves when attacked on it, according to party officials and strategists. A Democratic-leaning group, the Health Information Campaign, on Wednesday began a $2 million, three-week national cable TV and Web advertising campaign to highlight those changes.

“If you voted for health care and you’re attacked on it, you say, ‘OK, whose insurance are you going to take away? The kid with diabetes? Or my daughter who just graduated from college?’ And you win that debate,’’ said Democratic strategist Tad Devine.

That strategy is playing out in Ohio’s 16th District, where freshman Representative John Boccieri, a Democrat, is battling a challenge from Republican Jim Renacci. Boccieri opposed the health care overhaul on its first pass through Congress, then voted for the final version. At the time, President Obama praised Boccieri for his political courage.

Renacci, in an interview, said his opponent defied the will of his district and caved to pressure from House Speaker Nancy Pelosi. “People here believe he does not represent the district, he represents Nancy Pelosi,’’ said Renacci.

Boccieri said his campaign is about putting people back to work. His campaign website doesn’t list health care among key issues, but he said he’s ready to defend his vote. “I’m looking forward to standing shoulder to shoulder with my opponent and asking why he doesn’t want the people of the 16th District to have the same health care as he would get as a member of Congress.’’

But Democrats who are most out front on the issue are those who voted against the legislation. At least five of the 34 House Democrats who opposed the health care bill last spring have run television ads to tout their “no’’ votes, as a sign of independence from Obama and Pelosi.

The president, at a news conference yesterday, said such Democrats are trying to make their best case to win.

“We’re in a political season where every candidate out there has their own district, their own makeup, their own plan, their own message,’’ said Obama, who noted the political difficulty for incumbents when unemployment is high. “They’re going to be taking polls of what their particular constituents are saying, and trying to align with that oftentimes. That’s how political races work.’’

One of the health overhaul opponents, Representative Jason Altmire, a Democrat from Western Pennsylvania, has a TV ad complaining that “too many people in Congress just vote the party line.’’ But not Altmire, the ad asserts: “You saw it when he voted against health care.’’

Altmire’s conservative district includes affluent suburbs north of Pittsburgh and old steel towns. “When the health care debate was taking place in the spring you heard that this was going to be a powerful issue for [Democrats] in the fall,’’ said Altmire, in an interview. “We were going to go around the country, we were going to campaign on it. Well, that hasn’t happened.’’