I'm preparing a new website. Is cross-browser incompatibility really a problem?

I think it's sincerely a problem .
Most of the users & visitors really care about design , UI so if your website missed up , and the design not compatible with browsers (IE , FireFOX ,..etc.) this will make them leave.

Yes its matter that to have your website compatible with major browsers at least (Firefox, Chrome, Internet Explorer, Opera, and Safari).
Ofcourse that doesn't mean your website should look exactly the same in all browsers cause each browser render it in different way depending on its engine.
but its important that you have you website functional correctly in major browsers and the layout is not damaged.

yes it is a big problem sometimes i my css work perfectly on firefox but when i open it on IE, it doesn't but if you work with CSS3 you can add this code and it can give you the solution

It's a real pain in the neck, mate..
The real problem happens when each web browser interprets CSS properties in a different way..
Moreover, you'll need to adjust your javascript code to work properly and the same on each browser because the DOM (Document Object Model) varies from one to the other and that you can avoid by using JQuery which is great at handling Cross-Browser issues concerning the scripting part..
I hope this adds something.

It is an issue that you cannot afford to ignore.
Also, bear in mind that some users will be accessing your site with mobile devices as well.
I know this may sound like opening pandora's box, but if your stick to best practices, follow web standards, stay away from vendor-specific CSS properties, and try to validate your pages, you will do just fine! http://webstandardsgroup.org/ http://validator.w3.org/

of course you must insure that your site is compatible with different browsers, it will not be the same but it should look fine, acceptable and almost the same in different browsers