Do not exist as physically-independent entities, yes. That means you define "existence" as in "this rock exists."
They might be quite real (and even somewhat powerful) as entities populating one's subconscious mind. That is the position adopted, by the way, by some buddhist schools. In that sense, the gods might "exist" very well.
The programs running on your computer "exist" on the same level as the subconscious gods anyway.
EDIT: Are you also saying that the subconscious does not exist?
It's pretty funny having this sort of conversations with compsci people. They tend to be pretty hardcore and either-black-or-white on issues that do not lend themselves to this sort of treatment at all.
I have no opinion on the subconscious -- in fact I don't really know what it is. It may refer to certain mental processes and then I'd say it exists, but in my opinion it is a very general term. I think that psychology is still too early to say for sure what does and does not exist.
If I had an irrational fear of 1000 legged toad and it affected my behavior it doesn't mean that it exists. Now of course we get into all sorts of epistemological problems that aren't solved about our knowledge of reality, etc...