As long as you don't have a private Twitter feed, apps can consume your Twitter via RSS or the API, and either republish the entire thing, or extract selected tagged tweets.
Twitter to Yammer
For Yammer, simply enter your Twitter username into your account preferences, and then any tweet hashtagged #yam will be copied to your Yammer status. Yammer Integrates with Twitter gives some information. In the latest version of Yammer, the setting is under Account->Profile...Twitter Username.
Twitter to LinkedIn
In LinkedIn, Edit Profile->Twitter... lets you pull in all tweets, or only tweets hashtagged #in.
LinkedIn to Twitter
You can also push status updates from LinkedIn to Twitter. See LinkedIn Learning Center - Twitter for more info.
To share updates from LinkedIn to Twitter, check the box next to the Twitter icon on the LinkedIn home page. The first time you do this, Twitter will verify your account name and password. Whenever the Twitter box is checked, that update will publish to your Twitter feed.
Twitter to Facebook
You can add an application (with all the associated security and privacy risks of Facebook apps) called Selective Tweets, once configured you simply use hashtag #fb to get tweets into Facebook. Another option is http://twitter.com/about/resources/widgets/facebook - The Twitter app for Facebook.
Facebook to Twitter
If you want to send postings from Facebook TO Twitter, use http://www.facebook.com/twitter/ - you can choose which types of content you want to be posted.
Twitter to blog/website
If you want to display tweets in a window on a site or a blog, you can use http://twitter.com/about/resources/widgets
Blog to Twitter
There are lots of different ways, one is Feedburner Socialize. See Feedburner Socialize and stats.
Be mindful that if you want to send status messages TO Twitter, you have to give the source application write access to your tweet stream. That means if the source application is compromised, it can be used to post malicious information (e.g. malware links) to Twitter.
You can see what applications you have added, and what permissions they have, at http://twitter.com/settings/connections - note that Twitter permissions are all-or-nothing, in the sense that you don't get to choose what permissions a connected application has - if it requests read and write, you can only permit it entirely or deny it completely, you can't choose to grant it just read permissions.
There are also various desktop tools, such as TweetDeck, that support viewing and posting to multiple different types of status feeds simultaneously.
This topic is related to the issue of moving content around in an enterprise and consolidating notifications.
Jan 14, 2011 RSS and enterprise notification architecture