I think it depends on the episode. If you took one that featured only one artist or band, I think that those episodes might suffer because part of the episode was also, at least a little bit, about that particular music.
I've been thinking about the music issue quite a bit, because it seems as if there should a way to release
some
of the eps. Perhaps in "best of" packaging. Any ep where the original crime took place before the 1960s should theoretically be much easier to get clearance for. For example, I think that the oldest case was from 1919, surely most of it would be in the public domain now.

Eh. The way they integrated music from each specific era was a big part of the show's appeal for me. I probably wouldn't bother watching a version of it without the soundtrack.
Anyone here mentions Hotel California dies before the first line clears his lips.

I really wish I had them all taped, would love to do what you did. Wish it were still on TV.

Do what I do.
I actually recorded each episode off TV. They played the original music, so I just taped it, and now have the entire seven seasons on tape.
I plan to convert them to DVD at some stage. I prefer to do it this way, and I don't have a problem with it. I am not buying it off pirates, and there is no official set, so a bit of dedication ( I even cut out the ads)the cost of tapes (but then I save on buying a DVD box set)and a few months, and I did it.
I plan to do it for other series that don't have a box set release (e.g. The Wonder Years), but have to rely on when the TV stations show it next, and be ready and available to record each and every episode.
I buy box sets when available, but with shows like Cold Case, I do this and keep the music intact. It is a bit of work, but it is probably your best bet.