How do you know there are no problems? The problems won't be for you, they will be for the people trying to use your site. And if they struggle using it, they're far more likely to give up and go elsewhere than tell you your site's broken.
Limiting your site to 770px isn't the best solution, because it can look quite narrow on larger screens. In an ideal world, you'd have a flexible design that would stretch to at least 1000px, and shrink to 750px or less, to best fit the user's browser window. That way, pretty much everyone is happy.
Yes, you do need to take account of smaller devices. On my own site, nearly 10% of all users are on mobile phones or similar. While most 'full browser' phones have quite a good pan-and-zoom function, the wide the page, the harder it is to use effectively, if you have to keep scrolling left and right. So again, allowing the page to shrink to <800px will help people there.