The attacks and kidnappings appeared to be in retaliation for the deaths of protesters in clashes Friday, and are likely to further strain tensions between Iraq's minority Sunnis and the Shiite-led central government.
Also on Saturday, Iraqi lawmakers said parliament has approved a law that would limit the terms of the prime minister, president and parliament speaker to a maximum of two terms. The measure, which must still be approved by Iraq's president, could pose a challenge to Prime Minister Nouri al-Maliki's plans to seek the post again in 2014.
Fallujah police Maj. Rasheed al-Adeeli said one of the soldiers killed was hit by a sniper on the outskirts of the city. Another was shot dead when gunmen attacked a military post where soldiers were packing up their equipment on the northern edge of the city.
Gunmen ambushed the car of three off-duty soldiers on the outskirts of Fallujah and kidnapped them, according to state television and a provincial police official reached by The Associated Press. The official, who spoke on condition of anonymity because he was not authorized to release the information, said authorities have launched a manhunt for the missing men.
Iraqi soldiers opened fire Friday on stone-throwing demonstrators near Fallujah, killing at least five, according to local authorities. Two other soldiers were killed Friday in apparent payback for the protesters' deaths.
Sunnis angry over perceived second-class treatment and what they see as unfair policies targeting their sect began demonstrating last month in Anbar. The protests have since spread to other predominantly Sunni areas.
The protesters are demanding the release of detainees and the cancellation of a tough counterterrorism law and other policies they believe overwhelmingly target Sunnis. Many link their cause with the broader Arab Spring and are calling for the downfall of the government altogether.
International rights group Amnesty International pressed the Iraqi government to immediately investigate the protesters' deaths and make its findings public.
"Anyone found responsible for abuses—including anyone found to have used excessive force against protestors—must be brought to justice," said Ann Harrison, Amnesty's deputy program director for the region. She also urged authorities to ensure security forces are properly trained and equipped to deal with demonstrators in a way that respects human rights.
Iraqi state television aired a statement from the Defense Ministry on Friday evening saying it would investigate what happened in Fallujah. Repeated efforts to reach ministry officials directly were unsuccessful.
The United Nations envoy to Iraq, Martin Kobler, said the U.N. is increasingly concerned about the rising level of tension within the country. He said he would welcome a decision by the defense ministry to investigate the incident.
"We urge the government to exert self-restraint in law and order, and respect human rights," he said in an interview. "Our message to the demonstrators is to exercise their right to demonstrate peacefully."
Kobler also called on political leaders across Iraq to "sit together in a constructive dialogue."
Muhammed al-Khalidi, a Sunni lawmaker, said that 170 lawmakers voted in favor of the law limiting the terms on the prime minister and other top officials. He said lawmakers from al-Maliki's State of Law bloc boycotted the session.
Mushriq Naji, a lawmaker from the bloc headed by firebrand cleric Moqtada al-Sadr, confirmed the vote. He said the measure is needed "to prevent the return of the dictatorship to Iraq."
"This law ensures that no person can stay in power for an unlimited time and that democracy will continue in this country," Naji said.
Putting the law into force could prove challenging, however.
Iraq's president must sign off on the legislation. The current president, Jalal Talabani, is incapacitated following a stroke and is being treated in a German hospital, and it is unlikely he would be able to sign the law into force.
Legal expert Tariq Harb downplayed the significance of the vote. Even if it were to be enacted, the supreme court could rule that it contradicts the constitution and is therefore invalid, he said.
"This vote could be part of a political game because the Iraqi constitution does not limit the terms for the prime minister and the parliament speaker," he said. "Also, the same parliament could convene at any time in the future and decide to abolish the new law if the political map changes."
Yacoub reported from Baghdad. Associated Press writer Adam Schreck in Baghdad contributed reporting.