We do not use Flash. We have considered it, but nobody in-house had the time to learn how to create animations and design something we'd feel good about showing off.
I like the highly restrained use of Flash that I sometimes encounter on websites and would model any future use upon that. What I mean is small ads or banners that don't interfere (much) with the browsing experience.
I am often temporarily impressed by Flash-only websites, but they almost always disappoint in the long run... too much form over function.
BTW, this is not an exact duplicate thread, but some might be interested in reading
Do you flash?