Fukuoka District Meteorological Observatory said Aso received 19 inches of rain during a 24-hour period, while Kikuchi received 12 inches and Taketa got nearly 10 inches, Mainichi Daily News reported.
An evacuation order issued for the two prefectures has affected at least 68,000 people, officials said.
Police said the heavy rains were responsible for the deaths of at least five people and landslides buried at least two people.
Kumamoto prefecture reported at least eight homes were damaged or destroyed and floodwaters had swamped many buildings, Mainichi said.
"It's the first time I've seen the river overflow. The water is over three meters [about 9.8 feet] higher than usual," said Nobuyuki Nagano, 65, who lives near the flooded Shirakawa River in Kumamoto.
In the Taketa area, the Tamarai River overflowed its banks, washing away bridges, flooding homes and destabilizing power and water service, officials said.
The mayor of Taketa has asked that Self-Defense Forces personnel be deployed for rescue and aid operations.
The Japan Meteorological Agency urged caution, saying the rain was "like nothing that we have ever experienced."