The Telsur Project of William Labov and others examines a number of phonetic properties by which regional accents of the U.S. may be identified. The area that is most free of these regional properties is indicated on the map: eastern Nebraska (including Omaha and Lincoln), southern and central Iowa (including Des Moines), and northern Illinois (including Peoria and the Quad Cities but not the Chicago area). It may therefore be the case that the accents spoken in this region are deemed the most "neutral" by Americans. This is borne out in an article in the November 1998 issue of National Geographic Magazine, in which the locals' "neutral accents" are cited as one of the reasons why Omaha is home to a large number of telemarketing companies.
Notable media personalities from this region include former talk show host Johnny Carson, longtime NBC News anchor Tom Brokaw, and CNN Headline News personalities Chuck Roberts and Paula Zahn, both of whom were local news anchors in Omaha.
General American was based on the accent spoken in the Great Lakes region in the 1950s. Many midwestern and western accents closely approximate, but I woudn't go as far as saying that eastern Nebraska, etc has an accent closest to "General American". Most westerners and midwesterners tend to believe that they speak "General American". The narrowest definition is that: /u/ and /o/ are conservative, cot and caught are unmerged, pin and pen are unmerged, feel-fill are unmerged, &c., and it contains no stigmatized features as found in much of the South and the Northeast. Another definition is merely that it doesn't contain any *stigmatized* features, but can have the c-c merger. Yet another definition says that it can have the c-c merger, the pin-pen merger, the pull-pool merger, &c. Then of course there's also the argument about how sorry and/or tomorrow should be pronounced. For example, even people who live in the areas Lolly mentioned have variations on those two words. Also, other regions that consider themselves speakers of General American pronounce them differently. For example parts of Iowa pronounce sorry and tomorrow as with "-ar" and others pronounce it as "-or". Many Northerners pronounce both with "-or-". In the West there is also some variation: Southwesterners usually pronounce them as "-ar-", whearas most Northwesterners pronounce tomorrow as "tomoerrow", but yet sorry as "sahree".
As Guest points out, what is and isn't General American isn't always clearly defined. Thus it's really hard to say that "General American" is some monolithic entity which is pushing out other varieties when GenAm is vaguely defined in the first place.
Linguists have noted certain features considered GenAm (such as rhotacism) are indeed on the rise in areas formerly known more for nonrhotacism. However, other features which are not considered GenAm features (such as the "pin-pen" merger or the Northern Cities Vowel Shift) have clearly been demonstrated by linguists as having spread past their places of origin in recent decades.
It is safe to say that every native speaker of English in the US has some, probably many, features considered "GenAm" and some not. While we can note individual features spreading or receding it's not really the case that one homogeneous accent is taking over the country. In fact, linguists have generally found that accents have been diverging as time has gone on.
<<which tells me that Philadelphia English is dying out among younger natives.>>
That's interesting you bring up Philadelphia because just yesterday in the bookstore I came across a book by Labov which specifically dealt with hitherto unseen features which are spreading in Philadelphia amongst the young. And these are definitely non-GenAm features. Labov has been tracking these developments in Phillie English since the 70s and has seen them spread.
As I've said before, you cannot generalize that a particular accent is disappearing without looking at the whole picture. Labov is an expert in changes in Philadelphia English and has noted that many of its features are now less standard (and certainly less GenAm) than they were 20 years ago due to spread of new changes.
What I've said here doesn't just apply to Philadelpha, either. Similar patterns are emerging in other areas. As I said before, some GenAm features are spreading and some receding, but mountains of relevant linguistic research have demonstrated there is no such thing as a monolithic GenAm accent which is replacing all former ones.
I would recommend looking at the book Sociolinguistic Patterns by Labov for further information and details--it's a very interesting book.