Just about everywhere I have worked this has been an issue and during a recent 2 day offsite, this came up again and resulted in some passionate and vehement discussion. 

So whose job is it?

I think there is more to this than the kitchen. I believe when things like this are consistently raised they are a metaphor for something else going on in the organisation. 

I remember at another organisation I worked with people were saying that the culture would improve when the carpet was replaced. Maybe it would, and I believe the carpet was a metaphor.

Leaders should pay close attention to what could otherwise be described as trivial issues as they are indicators of wider problems.  Either way, your team members are making meaning from these.   

Dirty carpet - means you don't care
People not cleaning the Kitchen - we don't take responsibility
Poorly stocked kitchen or stationery supplies - think about finance over people

Car parking issues?
Sitting near windows?
Air conditioning?

What's the trivial issue on your workplace that could be a metaphor for a wider issue?