The protracted surge in bloodshed, with more than 3,000 people killed already this year, is among the long list of complaints, along with rampant corruption, high unemployment, and what government critics say is insufficient improvement in public services.
Preliminary election results are not expected for at least two weeks. Initial election commission figures said around 60 percent of 20 million eligible people had voted.
The turnout in 2010 was 62 percent.
- No government soon? -
As was the case previously, forming a government is likely to take months, but Maliki said on Thursday that he had the votes to put together a coalition.
"We have confidence that we will achieve a political majority," he told reporters in Baghdad's heavily fortified Green Zone.
"We have an ability to pass the 165 (seat threshold)" required to form a majority government.
However, he insisted he would not cling to the post: "My mother did not give birth to me as a minister or a prime minister. ... I am not interested in this subject (of being premier)."
While Maliki's bloc is tipped to win the most seats, no single party is expected to win a majority on its own, and Iraq's various political alliances and communal groups will have to form coalitions.
Complicating matters is the fact that the three main positions of power -- the president, typically a Kurd, the prime minister, normally a Shiite, and the speaker of parliament, usually a Sunni Arab -- are often negotiated as an encompassing package.
"Finding a balance between the three communities -- Sunnis, Shiites and Kurds -- is not that easy of a process," said Ayham Kamel, Middle East and North Africa Director for Eurasia Group.
Kamel predicted forming a government could take three to six months, noting: "It's a bit difficult to do all of these things at once."
Wednesday's election, the first since US troops withdrew in late 2011, came amid a surge in bloodshed, with new government figures showing more than 1,000 Iraqis killed last month alone.
Figures compiled by the United Nations and AFP also illustrated the continued spike in unrest.
On election day alone 14 people were killed, including two election workers.
- Rebuke to extremists -
But a security clampdown meant less violence than in the preceding two days when nearly 90 people died, with Washington and the United Nations hailing the vote as a rebuke to extremists trying to derail the political process.
"The people of Iraq know better than anyone else the enormous challenges that they face," US President Barack Obama said in a statement, "and yesterday's turnout demonstrated to the world that they seek to pursue a more stable and peaceful future through the political process."
UN Secretary-General Ban Ki Moon welcomed Iraqis' "determination to strengthen the country's democratic process", his spokesman said, adding that balloting had been carried out "in a professional manner under challenging security circumstances".
"The Secretary-General encourages patience while the ballots are counted and complaints adjudicated," he added.
Despite the myriad issues facing Iraqis, candidates largely appealed to voters on ethnic, communal or tribal grounds, and the campaign itself hinged on Maliki's bid for a third term.
His critics have accused him of concentrating power and marginalising the Sunni minority, and say public services have not sufficiently improved during his eight-year rule.
Maliki contends that the conflict in neighbouring Syria is fuelling violence and has accused Sunni Saudi Arabia and Qatar of backing insurgents.
Analysts had expressed fears many voters would stay home. But many Iraqis said they had voted despite the unrest because they were tired of their elected officials.
"I hope to change all the current politicians, especially members of parliament, because they are thieves and are looting the country's money," said 91-year-old Jawad Kamal al-Din, who hobbled to a polling centre in west Baghdad.