It is believed two primaries and a secondary school were issued with threats by the Red Hand Defenders on Friday.
Mr O'Dowd said the news was "disturbing".
He added: "Regardless of anyone's politics or views, schools should and must be havens for the young people who attend them and the staff who work there.
"As Education Minister my priority is to ensure that children are educated in a safe and secure environment, free from threat or intimidation.
"I believe this is a priority all our society should share."
He continued: "Threats such as this should be condemned by everyone in society - not just political, religious and community leaders, but all of us.
"Communities must unite against such threats.
"Whatever difficulties arise in our society, we cannot let them disrupt our schools and threaten the lives of children, young people and the staff who work there."
On Monday morning there was a small police presence at one of the schools involved.
It is understood that the schools are advising parents to allow their children to attend as normal but accompanied by parents or guardians.
The police have said they are aware of the threat and are investigating. They have urged community leaders to "play an active role in diffusing tensions".
A spokeswoman said: "At this stage there is no evidence to corroborate the threat.
"Police are committed to protecting all communities in north Belfast and we would encourage all community leaders and those in positions of influence to play an active role in diffusing tension and maintaining calm."
Meanwhile, local DUP Councillor Lee Reynolds has urged residents to treat the reports "with care", adding that "people are playing games".
"There are persistent attempts to increase tensions in north Belfast, often through false rumours and claims," he tweeted on Sunday night.
"The threats against the schools story is the latest example. Those behind it are to be condemned."
The Ulster Political Research Group and the Progressive Unionist Party (PUP) - which have links to the UDA and the UVF - also both condemned the threat.
The PUP has described the situation as a republican attempt to heighten tensions in the area.
Meanwhile, graffiti threatening Protestant workers in the FG Wilson (Caterpillar) plant at Springvale Business Park, off Springfield Road, Belfast.
The graffiti has since been removed and local republicans have said there was no community backing for it.
Greater Shankill DUP Alderman Frank McCoubrey condemned the appearance of the sectarian graffiti.
"This is an utterly disgraceful example of sectarian intimidation of Protestant workers in a mixed workforce," he said.
"Those responsible are seeking to cause division and to increase sectarian tensions, not just within the workforce but in the wider community. Their pathetic actions will be rejected by all right-thinking people and I call for clear and unequivocal condemnation from all political parties."
He said he has requested an urgent meeting with company managers and local PSNI representatives to enquire about protection for the workers.
There have also been reports that police have warned loyalists in the Twaddell Avenue peace camp of threats made against them by dissident republicans.
The camp was established at the north Belfast interface after the Parades Commission barred Orangemen from marching past the Ardoyne shops for this year's Twelfth.
Picture of the graffiti which was daubed on the side of a shop at Springfield Drive, on the lower Springfield Road (© DUP)