I have three Zenith DTT901 Digital TV Converter Boxes "deployed" in my house. (It's been a very good box and I'd recommend it if you can find one.) In two instances I'm using the remote that came with the unit. It handles a) control of the box, b) turning the TV on/off, and c) controlling the volume via the box instead TV. (I would prefer for the volume to be controlled by the TV, but it works fine as long as you "fix" the volume on the TV itself at the right point.)
With the third TV, I'm using an old Sony RM-VL600 universal remote since I have the TV, converter box, VCR, and DVD player all part of that setup. The RM-VL600 is a learning remote and I "taught" the buttons for use with the converter box. The remote itself did not have a code enabling it to be used with the converter box and neither did either of the other two "regular" universal remotes I had laying around. But all of these remotes were purchased before the converter boxes came out, so a new universal remote you purchase today may have that support. But "out of box" support will be something to look for.