Bin men are hard at work to clear rubbish in areas such as White Park
The council had already pledged to get to grips with the collections following the end of last month's snow falls, but some parts of the Capital were left waiting.
It led residents to accuse the council of providing a "gold-plated" service for parts of the city centre, while ignoring outlying areas.
Heavy snow falls and freezing temperatures saw refuse collections curtailed for most of December.
Keith Johnstone, a resident of Stenhouse Gardens, said communal bins had not been lifted in his street for well over a month.
He said: "We've had some bin collections, but the communal bins haven't been collected for over eight weeks now.
"I've phoned and e-mailed the council, but they have done nothing about it.
"I know the bin men have been on strike and the weather has been bad, but this is just ridiculous. You go to the New Town and they get a gold-plated service there."
Kevin McGuire, a resident of White Park in Gorgie, said there had been a similar situation there.
He said: "You couldn't get into the bin store, the bin bags were just piling up and overflowing into the car park.
"People had started keeping their rubbish in their homes, which attracts vermin. It's a disgusting sight."
The council said there would be no wheelie bin collections today, Tuesday or Wednesday, before things begin returning to normal. Temporary skips are also set to remain in place for the time being.
A council spokeswoman said much of the backlog had been cleared by the end of last week.
She said: "Christmas and New Year always mean a huge increase in the levels of waste to be collected, and collecting staff have been working extremely hard to clear the backlog.
"With snow and ice no longer hampering collections, refuse teams have been out in full force and are collecting in all areas of the city as normal."
Meanwhile, the council said Christmas trees could be recycled at any of the city's four community recycling centres.