I have experimented with Blender before making some simple low poly landscapes.
I found this example online:
The scene I am trying to animate is very similar to the example above but a low poly mountain range. How can I achieve this effect?
You can use a Displace Modifier and an Empty to move the coordinates of the Texture doing the displacing. This produces the "swimming through" effect. Use the Strength slider to adjust how subtle or pronounced you want the effect to be. Keyframe the Empty as needed to get the speed you want.
You can also change your Texture Type and Size to best fit your mesh density:
Rotating and Scaling the Empty produce interesting variations on this effect. Play with it. :-)
One more thing: Here's another way to do it with the Empty coordinates using a driver, so you don't even need to keyframe. If you type
#frame into an input field then the value will move with the frame. Then you can change it to frame/20, frame/30, etc. for finer increments (slower change). This method uses Python, so you can "Reload Trusted" or enable "Auto Run Python Scripts" in Preferences.