Cyanogenmod without google apps I flashed my Android with Cyanogenmod 11 without google apps, the sole intention being to avoid Google whatsoever. I am using Runtastic to log my runs, but it doesn't display the map. Understandbly, because Google maps app is not installed. My question is if I can install some other maps like bing maps etc. and use it with Runtastic.

PS: I have no intention to install Google maps or any other google product on my phone.
You might wish to take a look at the NOGAPPS project, which is what I am using. Without having to install the GApps, it e.g. provides a MapsAPI in a way that other apps think it's GMaps. There's also a NetworkLocation provider able to use a locally stored CellID database – found my location on the spot. Third part is the BlankStore, allowing access to the PlayStore including License-verification for paid apps (you cannot buy apps with this, but you can use apps you've already bought). Perfect trio.

You might also wish to see my article on this topic: Android without Google 6: Self-Experiment (need to update it with my progress on another device, where all 3 components work fine).