good news for you! and no, a cable with HDMI on one end & DVI on the other is fine. unless this is a bargain basement cable, 2-3 meters, it has good potential to work for you. there is no guarantee of success, tho.
if you get speckles, snow, lines and you aren't going to buy a new TV for a time, there are repeaters, boosters, even a switcher will help boost, for signal boosting.
if you get no video & the HDMI LED on the receiver is flashing, then there's a problem with the HDCP handshake. you can try another input for the source or try the other HDMI output. also check to see if the HDMI light is solid when just the source and receiver are connected while the receiver & TV are not. if it's solid then turn off the receiver, reconnect to the TV and see if it flashes on the receiver. if it does, then the TV may be too old for its identifier ID to be recognized by the receiver.
I encountered this myself with a Pioneer universal pre-Bluray player; it's HDMI 1.0 spec and the signal gets passed to the TV A-OK thru a switcher but not thru either Pioneer receiver I have (SC-09 & SC-68) it just doesn't make the handshake thru the receivers. I have a similar issue with a JVC D-VHS hi-def deck that's HDMI 1.1 - while I get connectivity, it only works for 480i not 1080i that's on D-Theater tape. and that also used to work thru the same switcher to my DVI TV. ended up using component video on the JVC and I don't use the Pio uni player for DVD's anyway, just hi-rez music so moot point. but I mention this because even tho HDMI is supposed to be backwards compatible...it really isn't
keep us posted on how it works out