Or, like Amazon.com (the worlds largest e-tailer), you don't use use things like flash that alienates the millions of people still on dial-up that will instantly leave your site when it takes longer than 20 seconds to load. Or not work correctly on the 18% of the market that isn't using MSIE and flash is either painfully slow (still a problem with some flash animations in firefox/mozilla) or not render correctly (Mac and Linux/Unix systems). I will not use MSIE on my windoze machine, and none of my other machines can even run it if I wanted to, so I see this a lot when visiting heavy flash sites (speedtv.com is a prime example ... their main page does not render correctly in Firefox).
A pretty website with animaitons often isn't the best way to do things
If everyone was on a broadband connection and you knew the pages would render exactly the same for everyone ... then, it wouldn't be a concern. A mixture generally isn't bad, but in many cases it doesn't really add much to the presentation unless you've got a graphics designer doing your animations for you.
EDIT: I have seen a few sites done entirely in flash that actually have a very good UI, load fast, and render correctly on all my machines. They are few and far between though, and you'd need a REALLY good flash designer and do a lot of testing to achieve it.