Thank you all!

That looks like the right solution. Now I have to build a frame with light sockets in the middle of each edge.

I'll be 3D printing for a while!

I have 100 LEDs on hand, individually controllable, with a Teensy computer chip to drive them. I'm going to need to step through the Graph3D so I can keep my place as I go!

@J. M. : This definitely is the ultimate solution (+1) - thanks for sharing!

... that's just what I wanted.[...] But now I'm wondering if there's a way to find a path that follows every edge?

Hmm - interesting problem! One simple approach might be to create points along the edges ...

lineEndPts = N[First /@ PolyhedronData["TruncatedIcosahedron", "Lines"]];
linePts = DeleteDuplicates@Flatten[Subdivide[#1, #2, 10] & @@@ lineEndPts, 1];
Graphics3D[Point[linePts]]

... and then calculate the shortest tour along these points:

{length, order} = FindShortestTour[linePts];
gr = PolyhedronData["TruncatedIcosahedron", "Graphics3D"];
Graphics3D[{List @@ gr, Red, Thick, Line[1.05 linePts[[order]]]}]

If you can live with the fact that some of the edges are not covered in one go, then this might serve as a solution.

Here is one way:

pts = N@PolyhedronData["TruncatedIcosahedron", "VertexCoordinates"];
{length, order} = FindShortestTour[pts];
gr = PolyhedronData["TruncatedIcosahedron", "Graphics3D"];
Graphics3D[{List @@ gr, Red, Arrow /@ Partition[1.05 pts[[order]], 2, 1]}]

Yes, this seems to be the better idea for this case, as - if vertices are numbered accordingly - it gives a plan on how to proceed:

graph = Graph[PolyhedronData["TruncatedIcosahedron", "Skeleton"], VertexLabels -> Automatic];
(* path from vertex 9 to vertex 11 - as an example: *)
hp = FindHamiltonianPath[graph, 9, 11];
HighlightGraph[graph, PathGraph[hp], EdgeShapeFunction -> ({Thickness[.02], Line[#]} &)]