At each vertex, the piece that has been removed is a regular tetrahedron.

Three of the edges of this are the pieces of edge that are lost from the original tetrahedron, the other three are the new edges of the truncated tetrahedron.

Since all the edges are the same length, the total length of the edges remains unchanged.

Therefore, the total length of the edges is $6\text{cm} \times 6 = 36\text{cm}$.