You can achieve this using the Skin modifier as described in How to 'skin' an edge.
Start with a subdivided cube :
Add the Skin modifier. In Edit mode, select all the edges and adjust the Mean Radius for each edge to affect the distance (use the Ctrl+A sortcut (while in Edit mode) or adjust in the properties side-panel) :
Add a Subdivision Surface modifier to smooth the surface :
Use the Skin modifier Smooth Shading checkbox to affect shading of the resultant mesh.
Note that you don't need to delete the faces since the Skin modifier only uses the edges of the original mesh.