- cross browser comp - Flash is a plug-in that is browser-independent. The broswer, any browser, either has the plug-in or it doesn't, and if it doesn't, you display alternate content. See tedster's reference to SWFobject.
In fact, take for example the puzzle of displaying a "sample sound" on a page. Other technologies that do this require a specific plugin less "universal" than Flash, or definite browser-specific approaches. Flash negates all of these problems by putting the sound in Flash, actually resolving cross browser problems.
- pop up blockers - Flash is not a pop-up, and pop-up blockers won't affect Flash . . . unless it's in a pop-up. :-)
- bad seo - only bad if you use Flash in a critical site function (using tedster's example of navigation) or critical content, as in, putting your text in a scrollable Flash object. Even this is changing, search here for thread on Flash and Indexing (as in search engine indexing) - they've been reading Flash content for a while, and Adobe has added some things to Flash to "feed" the search engines for even more content.
In the context of an animation, header, supplemental info that is otherwise difficult to access, Flash is just what it's supposed to be - an enhancement of your page.
The point, take advice with a grain of salt until you investigate on your own, it may be generalized, interpreted incorrectly, or even incorrect (even mine! :-) )