Using management studio, right click on the dimension and choose "script dimension as" -> "create to" -> "file".

If you're using source control, you can check the XMLA file in.

Manual deployment is easy - just connect to your new database, open the XMLA file, and execute it.

Scripted deployment is possible also. You need ascmd, which is a micofsoft tool similar to sqlcmd that lets you run XMLA files from the command line. You have to download it - it's not on the site I originally downloaded it from but I've heard it's part of Project Real.

Also, see the readme for ascmd.