How was the shower waterproofed? Have you kept it sealed?

Travertine is porous. Is it possible you are just growing mold/mildew on/in the pores of the travertine itself?

We have kept the shower sealed. I think the tile man did a faulty install. It was a travesty. The workman actually installing the tile brought the saw into the house without our knowing it until it was too late & we had dust throughout the house.

Most travertines(darker travertines can be porous) are not very porous-the material is dense and is formed in calcite rich springs underground. Because of the natural holes and pitting its easy to think it may be porous.
It is a stone that works well in my opinion in wet environments. Provided that the install was done properly.
Are all the joints(corners,base,etc) properly caulked.
Is the pan clogged and holding water. To make sure it is mold test it with full strength bleach. Just a small area-(yes bleach- dont worry it wont hurt the stone) It will if it is mold or mildew remove it. Post a picture if you can. More likely than not there is a cure for your shower stall.