UPDATE 2 - 6/21/2017: Most all of the below methods no longer work, especially on devices with Lollipop or higher Android. Currently the only effective method of hiding root access is using Magisk to root the device and use Magisk Manager to enable the Magisk Hide module and hide root from the app in question. Since this is an old question and I am just updating an answer, here is an article describing how to do it and below is a pretty rough guide to doing it.
Summary of steps:
- Remove old root method complete, this may mean you need to restore
stock or reflash your ROM. Some ROMs have root built in, so check
with your ROM maintainer/developer.
- Flash the Magisk installer ZIP using TWRP or other custom recovery
- Download and install Magisk Manager
- Swipe from left and go to Settings and enable Magisk Hide
- Press Back and select Magisk Hide
- Scroll through the Magisk Hide menu and select the applications you
wish to hide root from
Your app should no longer detect that you have root.
Old answer - May still be applicable to devices with KitKat and previous Android versions
UPDATE - 9/2016: Most of the methods below will now fail since Google has updates SafetyNet API in how it detects root and modifications to a device. The SafetyNet API is a mandatory part of Google Play Services that auto updates itself regardless of your settings, and many apps are now using this to detect root or modified systems. In fact, in Android 7.x devices, an otherwise stock device with merely an unlocked bootloader can fail the SafetyNet check. More information on SafetyNet API can be found here.
RootCloak will work with most applications to hide root, it is an Xposed module, or RootCloak Plus which is a stand-alone app.
The reason banking and other financial apps don't work on a rooted device is for the potential of private information being obtained by other apps or users with root permissions. Like it or not, there is no denying that having a rooted devices can potentially have security and privacy concerns, even if only in theory. Suhide by Chainfire is another great solution.