This, like the heat distressing variation, requires a little planning ahead. You want to use only threads or fabrics that you can dye - and, if possible, thread that you can dye also. For this piece I layered a variety of smooth and textured whire rayon yarns for the Solvy sandwich. I didn't have any white silk sewing thread - and, as DharmaTrading points out, when you need it is a bad time to go looking for it - so I used white rayon thread and hoped for the best.
After sewing the sandwich and rinsing out the Solvy, I soaked the piece in a soda ash solution for a while, wrung it out very gently, and applied minute quantities of dye with an eyedropper. The Madeira rayon embroidery thread didn't take the dye; it must have some finish that made it resistant. The rest of the piece, though, dyed very nicely indeed.