Firstly, Flash requires the installation of a browser plug-in to display its content, but the browsers that support it are mostly limited to personal computers. So a flash heavy website will not display correctly for other devices with web browsers, such as cell phones, PDAs or some of the most popular gaming systems.
Secondly, it is still unknown how well Flash based content gets indexed in search engines. Traditionally, Flash based content was ignored by all search engines. Although Google and Yahoo have announced that they are able to index flash content, the SEO benefits and success have yet to be proven. Some common things we see from our clients are the use of Flash as their primary navigation source. If your primary navigation is contained within a Flash movie, then you need to consider an html based alternative. Probably the best use of Flash, in our opinion, is for rich banner ads, or aesthetic elements that do not rely heavily on important content.
Overall, if you are concerned about search engine rankings and you have a content rich website, it is best not to design an all Flash site. If you still want to go down that route, make sure you have an alternative html option, such as those listed above, so search engines will play well with your website. As of right now, it does not appear that you can have an all Flash website without some kind of supplemental solution. Only time will tell if Flash and SEO are truly good bedfellows.