Is it important to design for all the major browsers?
I think it is. You can't make people use another browser without seriously hurting your site's traffic, and some people just leave if your site doesn't look good or is broken in their browser.
The browsers I try to make my designs compatible with are Chrome, Firefox, Opera, and Internet Explorer (as much as I hate it).
By the way, if you design for Chrome there really is no need to install Apple's Safari, because they both use the same software underneath it all.
What do you think? I'd love to stick with Chrome and Firefox, or even Opera since it's pretty easy to make things work with it, but it's a real pain to get things working in IE.