A material such as this should be useful.
This combines a couple of Wave textures (at the same scale as the checker texture) to identify the edges - in this way you have a graduation as you get closer to the edge rather than a sharp edge. You can vary the Power node to bring the 'edge' closer to the edge of the Checker tiles (eg, 2000 would get it really close, 1 gives you a graduation over the whole tile).
I've not quite got it clear in my head yet why those values in the Location and Scale of the Mapping nodes - the scaling on the Wave textures is 1/4.5 and that seems to work fine (I'll try and figure out a more scientific reason for that at some point!). Obviously the Scale of both the Wave textures and Checker texture must match. Also, if it's different to 5.0 then you'll need to adjust the Location offset of the Checker texture to suit. I'll try and figure out a better way of getting these to line up but, for now, hopefully this will be useful.
I've combined it with the Checker texture with a Mix Shader just to show that they combine correctly - vary the Mix Factor to blend between the two.
Here's an example using this as a Bump map.
And here's the Blend
EDIT : Here's a "simplified" material using Cosine maths nodes in place of the Wave textures. This simplifies the scaling so that a single Value node can be used to set the scale for both the Checker texture and the Edges (the output of the Power node). Note that the second column of Multiply nodes are set to multiply by 'pi' (which blender kindly expands to the value of pi).
Blend file attached