Kenya has been the hardest hit with the government recording 194 deaths.
In Rwanda, 55 people have died and floods have killed 16 in Somalia. In Uganda high water levels have trapped an estimated 200 patients inside a hospital.
East African countries have also been hit by a locust invasion and Covid-19.
The authorities in Kenya have told people in some of the affected areas to move away from "potential danger".
The water has also washed away 8,000 acres of crops and some vital infrastructure, the government has said.
Meteorologists predict that the heavier than usual rains will continue throughout May, reports the BBC's Patience Atuhaire in Uganda's capital, Kampala. The rainy season normally lasts until June.
In Rwanda, people died as a result of landslides in the mountainous north-west of the country.
Houses, roads and crops were all destroyed and hundreds remain homeless, mayors of the affected districts have told the BBC's Great Lakes service.