The pilgrims were returning from the holy city of Kerbala, south of Baghdad, where millions gathered over the weekend for the Arbain ritual despite attacks by suspected Sunni Arab insurgents that have killed scores and raised communal tensions.
In eastern Baghdad, a suicide bomber blew himself up on a minibus, killing 10 people and wounding eight, police said. The blast ripped the minibus apart in an area of northeastern Baghdad near the Shi'ite militia stronghold of Sadr City.
In another attack, five construction workers were killed and 10 wounded when their bus hit a roadside bomb near Baladruz, 100 kilometres (60 miles) northeast of Baghdad, in the violent province of Diyala, police said.
Further north in Mosul, a bomb ripped through the office of the Islamic Party, killing four people, said police Brigadier General Mohammed al-Wagga.
Elsewhere eight more people were killed.
Prime Minister Nuri al-Maliki had urged regional and world powers at a conference in Baghdad on Saturday do all they could to help end sectarian violence which is threatening to plunge the country into all-out civil war and spread in the region.
Saturday's meeting was a rare opportunity for Washington and its adversaries Iran and Syria to discuss violence in Iraq.
Iran said on Sunday it backed any efforts to quell violence in Iraq and described the regional meeting as a "good step".
"We support any efforts that will bring Iraq out of its current problems ... and help the Iraqi security. Iran will be the first supporter of this plan," Iranian Foreign Ministry spokesman Mohammad Ali Hosseini said.
But while President Bush on Saturday ordered 4,400 more U.S. troops to be sent to Iraq on top of a force build-up he has already authorized, Iran called on Saturday for the withdrawal of U.S. forces on grounds they fueled violence.