Actually, migrating from Serendipity to Serendipity is covered in the FAQ:
http://s9y.org/11.html#A21.
However, the Windows -> Linux thing can be an interesting experience.
Most of what Serendipity does is HTML/HTTP, which has standardized on the forward-slash (/) for all operating systems. So images, links, embedded content, and anything else that a remote system has to access from your server should pose no problems.
Paths in the Serendipity code shouldn't pose any problem, either. PHP uses forward-slashes in its paths, too. The PHP is interpreted through the web server, though, so that may cause some trouble. Apache will take forward-slash and backslash interchangeably, so if you're using Apache on either server, you should be good to go.
The only likely problem is in the php.ini file. That uses an include_path variable that may be interpreted by the operating system. That's a server setup issue, though.
Of course, you could check your serendipity_config_local.php and change all the paths there to use forward-slash. There shouldn't be anything in it that references a drive letter.
Finally, if you've done something silly in any of your entries, like including a drive letter in a link, you should modify that by hand. You can do it directly in the database dump, since that should make it easier to find all instances.
And if you have any trouble, we'll still be here to help.
