Seeing sites that feature an up-to-date Twitter feed is becoming pretty standard these days. For a while I was getting a bit annoyed with them. I then figured out what was bugging me - here is a persons / company's professional twitter feed, as showing on the home page of their site and half the tweets are conversational ones between the site owner and someone else.
Of course Twitter is great as a quick, conversational medium, but you don't want that showing on your homepage. If nothing else, your site's visitors will have no context for that conversation and will feel alienated. They can also come across as perhaps a little too informal, and various Retweets, whilst useful, don't really say anything about you yourself.
What you really want on there is the interesting tweets - the links, the opinions, the thoughts, the activity - this is what will communicate your personality, your brand to your site's new users, the visitors that know nothing about you yet.