When re-developing a website it’s sometimes not easy (Or sensible) to keep the old urls. So you are faced with 301 redirecting all the old urls to the new urls, and sometimes it can take Google a while to crawl and register all the 301 redirects (Then update the SERPS) – Especially on Google Custom Search results.
A little tip to speed this up, is to do the following
1) Save your old Google XML sitemap (With all the old urls) to your computer
2) Make sure your new site has a Google XML sitemap
Now in webmaster tools submit your new sitemap, and upload the old XML site to your new website and submit the old XML sitemap url as well.
Doing this will trigger a crawl of all the new urls AND all the old urls, which in turn will get Google to register the 301 redirects on the old sitemap much quicker.
We did this on a fairly large site, and within a matter of days the SERPS had been updated and all our custom search page results were correct. It was A LOT quicker than when we last changed a site and 301 redirected all the pages without doing this.