The Widgetkit Twitter module is the neatest way to display tweets directly on your website. All displayed tweets get cached so they show up super fast, even if Twitter is not available.
Create a Twitter module in Joomla or a Twitter widget in WordPress and publish it in a module position. A lot of options are available to choose which tweets you want to show.
For example you can show only tweets from a specific user or tweets that contain a certain hash-tag or a specific word. All options can be combined. To filter by more than one word or user use a space between them. For example: