Can I use a CSS (or other) technique to round the corners of an image, or do I actually have to round the corners of the image in an image editor?
You can style a div with rounded corners and simply apply the image as the background.
<div style=" width:80px; height:80px; -moz-border-radius:8px; -webkit-border-radius:8px; border-radius:8px; background-image:url('http://gravatar.com/avatar/0c5517f0ac2d435fa0927672e3d4f40b.png');">
However, this only works in browsers that support rounded corners via CSS, so actually rounding your images is always the best way of ensuring the visual output.
I'd go with Gary's suggestion.