I have a website that is getting a couple thousands visits from some good keywords. I am going to be upgraded to a dedicated server. Will there be any problem with my search engine rankings?
The dedicated server itself will play no role, but the server location can play a role (if you are switching hosts). http://www.digitalpoint.com/tools/website-country/
No it shouldn't have any direct effect on your search engine rankings. Location of your server may be a factor but if anything your search engines could increase because spiders will have better access to your site with the dedicated resources.