Re: No problem here - but there is...
I've rooted and installed cyanogenmod on a S3(work) and S4(personal), root within cm is something you have to enable so I'm not sure how banking apps would interpret that, Barclays does not like root though not tried it with cm, Tesco is fine with root as long as you don't certain apps installed e.g. wifinspect.
The great thing about cm is you get to choose what google apps you install, I recommend pico (http://opengapps.org/) and if you need anything else install the individual apps.
One tip I can give you that caught me out is "google play services" as your battery will drain at a ridiculous rate till you go into privacy in the setting and stop it from waking up etc... oh and those privacy settings are great for stopping every tom dick and harry from reading your contacts/location/media/camera/microphone.
You'll be glad once you do it because you'll actually start to think your phone is a little more yours.