The raids struck areas around the town of Tal Tamr in Hasakeh province, the Syrian Observatory for Human Rights said, without giving information on possible casualties.
They followed the launch Monday of a new offensive by IS, which kidnapped the Assyrians as it seized territory around Tal Tamr.
The town remains under the control of Kurdish forces, but at least 10 surrounding villages have been seized by IS, along with the captives.
The Observatory said negotiations were ongoing "through mediators from Arab tribes and a member of the Assyrian community" to secure their release.
The IS offensive has killed at least 35 jihadists and 25 members of the Kurdish People's Protection Units (YPG) and of an Assyrian defence force, according to the Observatory.
The fighting and kidnappings prompted around 5,000 people to flee the countryside, with many seeking refuge in Qamishli, a large city in the province that is controlled by Kurdish and regime forces.
"We've received around 200 families who are being hosted in local homes," Jean Tolo, of Qamishli's Assyrian Organisation for Relief and Development, told AFP.
Others made their way to Hasakeh city, the provincial capital, describing harrowing journeys after fleeing in their pyjamas.
The United States and United Nations denounced the mass abduction of Christians -- the first of its kind in the country -- and demanded their release.
"ISIL's latest targeting of a religious minority is only further testament to its brutal and inhumane treatment of all those who disagree with its divisive goals and toxic beliefs," State Department spokeswoman Jen Psaki said, using another abbreviation for IS.
The UN refugee chief meanwhile warned that Syria's refugee crisis was approaching a "dangerous turning point" with nearly four million people facing worsening living conditions in exile.
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