Summary
I have given this topic much thought (seriously). My conclusion is that zombie stories are popular because they give a relevant, contemporary but intangible fear (loss of hope) a very simple, tangible solution: Kill zombie, find food, survive.

The problem: Real problems with no tangible solution
Today's outlook is radically different from what it was 50 years ago:
- Impending environmental disaster
- Increasing secularity with no hope of 'eternal salvation'
- Shattered dreams: Space program cancelled. No flying cars. Capitalism failing.
- Shift of power (economical and political) from west to east
- Failing wars

All these threats are more or less real but INTANGIBLE. There is nothing any single person can do to make a real change. We crave a problem we can solve. This leaves a hole to be filled by stories. Enter 'zombie apocalypse'.

What are the central zombie stories?
I find that I don't find just any zombie stories interesting. The stories that I find fascinating are the ones which, to my mind, really signify the recent 'zombie outbreak' in stories the last 10-15 years;
- '28 days later' and the sequel '28 weeks later'
- 'Dawn of the dead' (the remake)
- 'The Walking Dead' (both the comic and the show)
- 'World War Z' (the novel)
- 'Zombieland' (to a certain extent)
- 'Resident Evil' (the first movie and the first game)
- 'Left 4 Dead' (the game)
- 'The Road' (I would argue this is related, although clearly not a clear cut zombie flic)
- 'I am legend' (the movie with the 'alternative' ending which is true to the original story)
- 'Diary of the Dead' (BARELY!)

Why are they relevant to the problem?
These stories have the following key elements in common:

Immediate tangible problems: Society as we know it has ended. There is no longer a long term future to hope or worry about, only an immediate problem: Survival. Resources are scarce and there is a constant threat level. Clear goals and actions: Kill zombie. Find ammunition. Find food. Find shelter. Survive.

Group dynamics: Small group of identifiable indivdiuals forced to function under these circumstances.

Journey: A 'journey' is undertaken with a clear solution at the end: Safety. Reuniting with lost family. Warmer climate south. Turning off of the evil computer. Finding a cure. Escape.

Moral ambiguity on a basic level: Not political ideals, abstract governance models, ambigous enemies in wars that cannot be won. Instead: Ambiguous but clear questions on tangible level: Who is my friend? Who is my enemy? Is my zombie son a zombie or a son? Can I trust the stranger enough to help him?

In short: We are lookng for a tangible problem to solve. It doesn't come much more tangible than putting an axe in a zombie brain.