I recently added support for microposts on my site. They'll appear on the main page along with full articles – they just don't have titles.
Manton Reece has written some great stuff about microblogging. Probably better than anything I'll write about it. You should read it.
I've been on a Twitter sabbatical for a couple weeks1. I am enjoying running at about 170% capacity instead of 180%. It's nice. Occasionally I still have the urge to share a snippet of text or a link. Normally I would turn to Twitter.
Now, I just add a micropost to my own site. I don't have to think up a witty title. I just write a sentence or two, tap the action in Drafts that fires off a Pythonista script, and the new post appears on my site.
Even when2 I invevitibly turn Twitter back on, I think I'll keep doing this. I like owning my words, even if I'm not so delusional to think they matter.