Hey guys, it's been quite a mystery how to backup battlefield 3, I found a working solution. (test on Windows 7 64-bit)
Step 1 # Open regedit > go to  "HKEY_LOCAL_MACHINE\SOFTWARE\Wow6432Node\EA Games\Battlefield 3" > right click > Export > save as bf3.reg
Step 2 # Navigate to your origin games directory.
Step 3 # Copy the battefield 3 folder to your backup location.
Step 4 # Install Origin on the new PC.
Step 5 # Copy the backup back into your Origin games folder.
Step 6 # Run the registry file and add to registry.

Open Origin, it will say preparing for a while until the process completes checking the files.

example data on bf3.reg
Windows Registry Editor Version 5.00

[HKEY_LOCAL_MACHINE\SOFTWARE\Wow6432Node\EA Games]

[HKEY_LOCAL_MACHINE\SOFTWARE\Wow6432Node\EA Games\Battlefield 3]
"GDFBinary"="C:\\Program Files (x86)\\Origin Games\\Battlefield 3\\GDFBinary_en_US.dll"
"GameExplorer"="{16DD7F7E-2EE1-4BD6-8008-FC2DB292220F}"
"DisplayName"="Battlefield 3™"
"Locale"="en_US"
"Product GUID"="{78225C16-4123-488A-BCE3-C83DC933C9CF}"
"Install Dir"="C:\\Program Files (x86)\\Origin Games\\Battlefield 3\\"

original posting and credit to BF3 Backup