The tools below all claim to be different from Twitter in one way or another, it is worthwhile to research them before deciding to start a microblog.
ShareTronix: Enterprise targeted social network where co-workers, clients and partners can share information, it can be used to assign jobs to people and security policy is configurable.
Identi.ca: Stream oriented social network running on open source (StatusNet) with 140 characters message limit, you can send private messages and create private groups, with group members needing administrator approval.
Plerb: Small microblogging tool where you can post pictures directly using a mobile device, the whole site feels like a Twitter clone, one of the main difference is that when you post a picture it appears on your feed and users are not redirected elsewhere.
Plurk: Service to showcase what you do in your life with a 140 characters limit, you can share links to videos and images with Plurk automatically creating a thumbnail view, a widget can be embedded into your blog.
CoopApp: Twitter like feed targeted at businesses, to communicate with other coworkers without distractions. It can be used to share daily agendas and coordinate meetings.
Chinese users will want to use a Twitter clone that is not censored and blocked by the Chinese Government, like Weibo(微博), and the most adventurous might want to host their own microblogging service using open source software, StableNet or Pump.io but be aware that spam can be a problem and it needs maintenance.