I first heard it by (can't think of his name--he's sort of like Tom Jones and Englebert Humperdink), and heard "that means". I would have bet that's what Hank sang, too, but I just listened and it does sound like "like me". Of course, in Hank's twang it more like "lak meee", which probably makes it easier to hear "that means" if that's what's already in your mind.

However, I always heard, and Hank clearly sings, "a robin weep", not "a robin grieve".

But back to Hank singing "like me", it reminds me of hearing Minnie Pearl saying that after singing "I Saw the Light", Hank told her he didn't see any light. Apparently he was pretty despondent, and these were little clues. In retrospect his suicide, intentional or just careless, doesn't seem too surprising.