The New Blogger has a simple way of adding snippets of your latest comments to your sidebar - it does by utilizing your comments feed.
To implement this go to your blogs 'Layout' which is accessible from your dashboard. Click on the Template Tab and then the 'Add New Page Element' tab. In the window that pops up choose to add add a new 'Feed' When prompted for the url enter the following address, changing the 'your-blog' part to the address of your blog :
http://your-blog/feeds/comments/fullWhen you are adding the page element you can choose how many comments you wish to display.
As an example, the address to add the latest comments snippets for this blog is:
Here is what this would look like on my blog:
However, when I was browsing the Beautiful Beta blog, I discovered yet another way they have found of improving the inbuilt method. As you can see from the image below, more detail is provided: post title, snippet of the comment and a read more link. I am using this method in my own sidebar.
Using this option requires you to edit your blogs template coding, so it is a little more complicated. The instructions and coding for you to cut and paste are to be found on the Beautiful Beta post New Recent Comments Hack.