Forecasts suggest two to three weeks of rain has fallen in just 24 hours.
The Met Office said this has now extended the weather warning until 9pm as the rain is now “a little slower to ease than earlier expectations”.
In Angus, all schools in Brechin, Montrose and Kirriemuir were closed due to localised flooding, as well as more than a dozen others.
On Friday morning, a lane of the M9 westbound was closed at J2 due to flooding. In Fife the A92 was closed between the Parbroath Crossroads and Melville Lodges due to flooding, and in Dunfermline, Halbeath Road was shut with diversions in place.
Regional flood warnings are in place from the eastern Borders to Aberdeenshire, with the Scottish Environment Protection Agency warning Aberdeenshire and Angus could be particularly affected.
There were 20 flood warnings and nine flood alerts in place on Friday morning.
Transport minister Jenny Gilruth said: “The Met Office is warning us to expect a period of difficult weather in parts of eastern Scotland throughout Friday, with heavy rain likely to impact travel in the amber warning area in the north east.
“The conditions could potentially bring disruption to the transport network, so it’s important people plan their journeys before they set off, particularly if they’re looking to use the trunk roads or travel by rail.”
Forecasters said some areas of higher ground in the amber warning area could see a month’s worth of rain within a couple of days, with more than 100mm of rain, and even 150mm, possible in parts of the Grampians and Cairngorms.
Average rainfall for November in Aberdeenshire is 113mm, while for Angus it is 122.8mm, meaning if these areas do see 150mm of rain, more than a month’s worth of rain will fall in the space of just a couple of days.
ScotRail has imposed some speed restrictions on lines in the east of Scotland until midday.
Network Rail Scotland said journeys would take longer today on the East Coast Mainline between Edinburgh and the Borders, as well as between Aberdeen, Inverness and the Central Belt because of the “extremely high rainfall”, which means train speeds need to be limited for safety.
Ms Gilruth said: “Motorists should make sure their routes are available, follow travel advice from Police Scotland and drive to the conditions.”
Chief Superintendent Louise Blakelock, head of road policing, said: “As an amber warning for heavy rain has been issued, we are urging motorists to plan ahead and avoid unnecessary travel.
“The Met Office, Sepa and Traffic Scotland websites have valuable information about weather disruptions and we would advise people to check these sites before setting off on your journey."