Redesigning a site won't affect your ranking in the search engine results pages (SERPs) if you do the following:

1) Keep the same domain name.
2) Use 301 redirects to your new pages if you have renamed them or added content. (Example: your old page was at "www.example.com/what-we-do" and your new one is "www.example.com/services".)

To do a 301 redirect on a Linux server you need to edit your .htaccess file in the root directory where you site is hosted. The simplest redirects, just pointing an old page to a new one, look like this:
"Redirect 301 /oldfile.html /newfile.html"
(with no quotes and one redirect per line)