Turkish Cypriot media said the car the two men and one woman were travelling in was swept off a mountain road near the town of Kyrenia on the ethnically divided Mediterranean island.
Authorities were looking for a missing fourth person in the same car that was found buried in mud a few hundred meters (yards) away.
Torrential rains over the past four days have pounded the small, eastern Mediterranean island nation, causing widespread flooding and power blackouts.
Some Turkish Cypriot officials blamed poor urban planning and construction in river beds for raising the risk of flooding.
Cyprus President Nicos Anastasiades, a Greek Cypriot, spoke to Turkish Cypriot leader Mustafa Akinci late Thursday to express his condolences and to offer any assistance. Akinci thanked Anastasiades and said Turkish Cypriot authorities were handling the search.
With the weather service forecasting more rain, Turkish Cypriot authorities ordered that all schools stay closed for a second day on Friday to avoid problems with possible flooding.
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