You have to download the latest DirectX 9.0c installer from Microsoft (dxwebsetup.exe), save it to a local directory, and then run it.
http://www.microsoft.com/downloads/...38-db71-4c1b-bc6a-9b6652cd92a3&DisplayLang=en
Also, in the Steam client use the option to Verify Game Files or whatever Steam call it, see if it finds anything to fix.
If that doesn't help, please run DXDIAG, Save All Info, and post dxdiag.txt here (first delete everything after the Sound Devices section so it will fit). Also post your Rome/settings.txt and Rome/logs/system.log files. Note that if you use Vista/Win7 and have installed the game in the suggested folder
c:\Program Files\Paradox Interactive\Rome\, the active versions of these last two files will have been redirected by Vista/win7 to a non-system directory, like
c:\Users\[name]\AppData\Local\VirtualStore\Program Files\Paradox Interactive\Europa Universalis - Rome\, where [name] is your Vista/Win7 user name. You may have to search your C: drive for them.
Also please open up the Rome /logs/setup.log file and search for "checksum" and post just that line here, thank you.