Watch the Open String Pedals online guitar lesson by Angus Clark from Hard Rock Survival Guide: Rhythm
We can use the open string as the static voice. There's many a great hard rock song that features fretted double stops, chords, or triads moving vs. an open string. You'll see this everywhere across the decades, from Zeppelin onwards.
Over an A, so many great riffs, from a more mid tempo thing like "Headin' Out The The Highway" or "Voodoo" by Sabbath, which both use 4ths over the A, to triads over a fast double time gallop like in "Crazy Train", to 5ths, where I think King's X have a couple of examples.
Over the D string, you've got a much lighter sound, unless you're in drop D. I usually think of moving chords over the open D as more prog, or like the bridge in a Sabbath song. This gets back to your triad exercises, so don't shirk them! There are just as many examples of 4ths, 5ths, thirds, and triads moving over the open E chord. This is really where the guitar shines in the hard rock context. Rock that open string!