So your organization has a Twitter account - but it's over there on Twitter.com. Just like your organization's blog, you need to integrate your Twitter feed with your website. If your organization uses a content management system (CMS), it is likely that an RSS/Newsfeed display module is included.
If you are not using a CMS or yours does not have an RSS module included, you can still integrate your Twitter (or other) newsfeed into your website. There are several ways to do that, and it gets easier all of the time. One way is to use something like tweet!, a jQuery plugin graciously provided by seaofclouds. Another approach is to use a widget.
Below we compare the tweet! jQuery plugin to Google's Feedburner widget. The jQuery plugin can be customized (colors, fonts, height, width, number of tweets, etc.). Feedburner is less customizable, athough it can be fed into other widgets, such as WidgetBox, where the look and feel can be customized (although we had a WidgetBox example set up, but it has ceased to work and we are not sure why and may investigate at some point in the future).