No, you just need an HD Box. We have an HDMI chord, and it is just a myth. We can not tell a difference on our 1080 Samsung LCD, though some people swear by them.
Actually, the HDMI chord really helps my one of my friends 720 HDTV, but Ive only seen an HDMI chord work well once.
but without a box, an HDMI chord will do nothing for HD