Hdmi to dvi-d back to Hdmi

I have a dvi-d cable run through the wall up to my tv and now need to replace that tv. My question is, can I buy an hdmi to dvi-d converter for the output of my cable box and then another dvi-d to hdmi converter for the input of my tv and if doing so, will sound be carried through the dvi cable to the tv? The existing dvi cable is already in the wall and it would be difficult to run another cable so I was hoping that buying two converters would be easier than trying to fish another cable but I've read conflicitin info about whether sound would be carried in the cable.