Twitter was a lot better before they decided that chronological order of posts wasn't the best order.
If you use Tweetdeck, you can set up a list of Nats Twitter users, and have a column just for them. It's a more passive experience, but usually more informative if you are following Nats writers, bloggers and reporters.
I use it a lot on gamedays when the GDT has devolved.
Uggghhhhh I do NOT understand how people use Twitter to converse during a game, or at all really. Trying to follow a conversation on twitter makes my head want to explode, and for as much as I love facebook it's just not the same as a messageboard when watching a game. Does twitter make anyone else crazy when they're trying to follow along??