The Batman: Arkham City Forums have word that they've tracked down some performance issues in Rocksteady's action/stealth sequel to DirectX 11 problems, and until a title update to remedy this is released, they recommend reverting to DirectX 9 (thanks PC Perspective). Here's word:

PC DirectX 11 Issues Please Read
We have received reports of performance issues from players of Batman: Arkham City on PC. After researching the matter, we found that running the game with DX 11 is causing the performance issues. We’re working on a title update to address this matter and expect to make it available in the near future.

In the meantime, a workaround for this issue is to run the game with DX 9 instead of DX 11. Instructions on how to turn off DX 11 are listed below.

We sincerely apologize for any inconvenience with your gameplay experience and thank you all for your patience as we work to resolve this issue.

Instructions for changing DX 11 to DX 9
The Launcher can be accessed on Games for Windows LIVE and Steam to change DX 11 to DX 9 in the Settings menu.
Change both DirectX 11 Features and DirectX 11 Tessellation to OFF.

Games for Windows LIVE
Start > All Programs > Games > Right click on Batman: Arkham City and select Launcher > Settings
1. Open the Start Menu
2. Select All Programs and then the Games folder
3. Right click on Batman: Arkham City and select Launcher
4. Choose Settings then click on the arrow next to DX options to change

Steam:
Launch Steam Client > Library > Right click on Batman: Arkham City > Configure PC Options
1. Open Steam
2. Select Library
3. Right click on Batman: Arkham City and select Configure PC Options
4. Choose Settings then click on the arrow next to DX options to change