As was mentioned this is caused by getting reduced to 8bits. You can get better looking maps if you use gimp to convert the original 16bit png image to .dds format. The loader will read it in as 16bit then.
See: https://blenderartists.org/forum/showthread.php?154090-DDS-Texture-compression-resources&highlight=dds
and then install the gimp DDS exporter linked to this wiki page,
https://jmonkeyengine.github.io/wiki/jme3/advanced/free_skymaps.html#gimp
See also Is PNG suitable for textures? for more info on why its happening.
You will still get the stepping you have but it is greatly reduced. You can then use
heightmap.smooth(1f, 2);
in your code to get even better results.
For better results creating your heightmaps in blender use the displacement bake option as is explained here,
https://blenderartists.org/forum/showthread.php?298106-Linear-grayscale-heightmap-from-plane-mesh&p=2409874&viewfull=1#post2409874 (link provided by Ali)
Adjusting the height of the BakeToPlane modifies the results.
I plan on making a tutorial eventually. Just busy currently.