Gary Fitzpatrick says his bins have not been emptied for almost a month.
Complaints have poured in from across the city after refuse collections were stalled by the big freeze.
Bin bosses now say they will "endeavour" to get to all areas that have not had any service in recent weeks by the close of play on Christmas Eve, amid warnings that a build-up of festive rubbish over the holidays could create a health hazard.
Extra staff have been drafted in to help clear the backlog, as one opposition councillor said the city council's reputation was on the line.
The worst affected areas include streets in Colinton, Balerno, Currie, and Saughton, and the council has warned that there could be "extended disruption" to refuse collections if the bad weather continues.
With no thaw forecast for the rest of the week and temperatures due to dip to -15C in parts of the Lothians tonight, concerns were growing over the backlog.
Labour councillor for the Pentland Hills ward, Ricky Henderson, said: "The real concern that people have been expressing is that it has been four or five weeks in some cases since they have had their bins collected, and as the refuse starts to accumulate with Christmas approaching, people are starting to get seriously worried about rubbish lying outside. The idea is quite horrendous at this time of year.
"It's good that the council have recognised that. I'm taking the commitment they made at face value and expecting every effort to be made. Their reputation is on the line and if they can deliver and fulfil that commitment, it would be a great credit to them. If they fall short of that, people will obviously be very disappointed."
The council said it was using smaller vehicles and refuse collectors on foot to try to reach areas inaccessible to bin lorries, with staff taking rubbish from wheelie bins by hand and carrying it to the vehicles.
Householders are being asked to ensure they bag their rubbish before binning it to make the job easier.
Around 70 temporary skips have been put out in supermarket car parks, schools and leisure centres for people to dispose of their rubbish if it does not get collected.
The council says it is prioritising its snow clearing on main routes, with 400 staff and sub-contractors, 22 gritters, 16 mini-tractors and 32 other vehicles involved.
Gary Fitzpatrick, 47, who lives in Saughton Mains Drive, said: "It will be exactly a month come Thursday since they emptied the bins. The streets are a mess, there's rubbish lying around the place."
Councillor Jeremy Balfour, leader of the Conservative group on the city council, said: "There is concern that there will be hygiene problems related to this, and after Saturday you will have things like turkey and other foods left over and yet we don't know what will be done with them.
"Collections seem to be patchy across the city."
Environment leader Councillor Robert Aldridge said: "We want to reassure residents that we are out on the ground, and are as prepared as we have ever been for whatever cold weather may come our way this week.
"We regret that we will need to be prepared for extended disruption to the refuse service if the forecast continues to be for further bad weather and will be looking at all possible options for dealing with waste."