There are pockets of information avaiable on migration strategies, but it's difficult to find concise instructions for migrating a content database.  By the way, I would recommend this approach over in-place or gradual simply because I don't lose sleep because it has the least risk of de-stabilizing the production SharePoint environment. 

If you are lucky enough to have a shiny new MOSS farm and you are trying to migrate the content to, then you are in the right post.  Please keep in mind that these are the simple instructions with superflous details purposely omitted.  I encourage you to research the details around each of these steps before making this your migration strategy.  You will find a few links at the end of this post that are comprehensive.  Also, I am only migrating the content database in this scenario, not the search [site]_SERV, profiles / audiences database [site]_PROF or single sign on [site]_SSO databases.  You can migrate those too by the way!

1. Download the upgrade pre-scan tool to your current production SharePoint 2003 server.  Here's the link http://www.microsoft.com/downloads/details.aspx?familyid=e8a00b1f-6f45-42cd-8e56-e62c20feb2f1&displaylang=en. 
2. If you're paranoid like me, backup your SharePoint 2003 farm using the SharePoint Backup and Restore tool.
3. Run the pre-scan tool from your SharePoint 2003 server using the /all and /c PREUPGRADESCANCONFIG.XML.  (The XML file tells the prescan the portal templates are not custom).  The pre-scan tool will identify any issues that will cause upgrade failure; and before you ask, yes you DO need to do this step.  It will also make some harmless modifications to your SharePoint 2003 databases.  If you encounter any errors in the pre-scan tool, view the log file and correct them.
4. Backup the SharePoint 2003 content database(s)--the [name]_SITE database(s) using Sql Server tools (2000 or 2005). I realize you already did this in step 2, but do it again anyway.
5. Copy the backup file to your MOSS database server and restore it using Sql Server tools.  This would be a good time to give it a name consistent with your shiny new MOSS databases.
6. Create a web application, but not a root site collection
7. Run the following command: stsadm -o addcontentdb -url [site collection url] -databasename [name of database you restored].  This may take several hours to complete.
8. Check out the migrated content by browsing to [site application url] and get a feel for what customizations are broken or missing.

Here are some very good comprehensive posts on content migration.  Please let me know if there are others.

MS-Advantage's SPS to MOSS 2007 Upgrade Tips (Database Migration) by Alan Coulter

WSS 3.0 & Sharepoint 2007 Database Migration Strategy Steps For Migrating A Sharepoint 2003 Site by Jason Fortner

Don't be afraid of Prescan - Part 1 by Joel Oleson

Your Friend Prescan.exe - How to Get it & What it Does - Part 2 by Joel Oleson

Deploy a new farm, then migrate databases by Microsoft