I found this answer by Guillo_Ortiz over in this thread. It solved the problem for me.
“When exporting the fbx, you get a file [modelname.fbx] and a separate folder [/modelname] containing its textures. I manually placed the fbx into its texture folder and then imported.”
In other words, put your fbx files in the same folder as the textures, and then import from there. Hope it helps!