Apps that are installed and available to the notification system on the smartphone are generally available to deliver notifications to the Microsoft Band through the built-in Notification tile, configurable in the Microsoft Health app. This works great. However, there is a drawback because if you use a Twitter app other than the default (I use Tweetium and love it), those notifications will be mixed in with everything else.
A big update to the Tweet It! app this week has brought Microsoft Band-specific support. When you enable the Microsoft Band option, a Tweet It! tile will be installed on the Band and all Tweet It! notifications will show up there instead of being mixed in with all other alerts. If sifting through notifications is an annoyance to you, you might want to check Tweet It! out.
Tweet It! for Windows is available for both Windows PC and Windows mobile.
Download: Tweet It! for Windows