The Sharingan isn't exactly needed for Chidori, but it is a big help. Since you must have an insane speed when using Chidori, you're eyes become completely focused ahead of you, and you can't see anything but what you're running at. However, the Sharingan lets the user take in all that the eye can normally see. This is demonstrated in the Naruto vs. Sasuke fight, in which Sasuke (with the newly acquired third tomoe to his Sharingan) could see Naruto's movements (which were dramatically increased, due to the Kyuubi's power) when he couldn't before with only two tomoes.
Wikipedia link on Chidori.
Originally Posted by Wikipedia Article on Chidori
This jutsu has a major drawback, however. The speed at which the attack must be done, combined with the fact that the user must run in a straight line, causes a tunnel vision-like effect for the user. This allows their enemy to easily counter the attack, making it a potentially lethal move for the user. Though this drawback would seem to make learning the technique fairly pointless, Sasuke and Kakashi are able to avoid the drawback because of their Sharingan eyes, which take in every detail regardless of how fast the user is moving.