Let me say for the record, I think Flash is a great plug-in. Lighter than Java applets, more secure than anything Microsoft tried to do, and so easy to install it ended up on over 95% of desktop computers. The development tools make it really easy to do cool animation and program logic, and I've seen some amazing stuff done using Flash.
But don't build your whole website around it.
The biggest problem I have with an entire site being built in Flash is that you can't bookmark pages, and in this age of link sharing through social media, that can be a site killer. ("Okay, go to www.bigassflashsite.com, click on the news link, look for the recent videos link, and scroll down to the one posted three months ago, then click..." as opposed to "go to http://tinyurl.com/xxxx.")
But this is an implementation issue, like so many problems I have with Flash driven websites. Flash sites are more likely to have that annoying intro, so annoying that it's become standard to put "Skip Intro" on the home page -- why, oh dear lord, why, would you make the FIRST thing someone sees something that you KNOW they are going to want to skip? Get me to the meat right off the bat, and tease me with your dancing robots after I'm hooked.
This goes for the long, complex transitions from one screen to another. If I have to watch a "Loading......" animation just to click on your Contact Us page, precious moments are slipping away where I might give up, get a phone call from your competitor, or decide I have to go to the bathroom.
The Web is an amazing tool for distributing information. Don't let the shiny objects blind your audience.