It may not be the hdmi cord. It could be the settings. Have you tried running the set up for that device? Just because the work on one, doesn't mean it will work with all. Every device is different, so you will probably need to do a complete set up for the new device to work with the TV. There also may be other changes needing to be done on the TV itself to make it work. I would check the website for the TV (you don't mention the brand) and see what they suggest for the new computer, vs the old one.
Also, in the case of the cord (if it isn't the setup) the older cord may not be enough for the new device and TV. Some older cords just don't work on newer systems. Odd but true. I would try the above first though.