It depends on what you're looking for. For instance, I don't particularly care about North American-centric anime news, so I tend to ignore it until someone tells me something interesting (more on this later).
I don't read ANN for the same reason I don't read particular newspapers; the reviews and editorials make me facepalm and I can get the news elsewhere. Since I use an RSS reader to keep track of my news, centralization isn't a huge problem for me, so that means that I have a ton of sources that cover specific news.
The site that is most like ANN in terms of breadth is probably
Japanator, so you'll probably be covered pretty well if you just follow them. Personally, I prefer the
MyAnimeList news board just because it's more specific in what kinds of news it covers.
For more obscure news, often involving 2ch shenanigans, I really liked
Canned Dogs when zepy was posting more regularly and
welcome datacomp. For seiyuu stuff, I like
hashihime and
Super Noisy Rice Shower Goes Boom!!. For figures, I'm just interested in Nendoroids, so I just follow
ミカタンブログ. For Japanese gaming,
Siliconera is pretty good. For visual novel news, I follow
encubed. And for new anime announcements,
moetron is excellent.
But the majority of my interesting news actually comes from my vast information network on twitter and google reader.