The answer to my question turned out to be rather trivial (at least for my rooted HTC Incredible). Trivial to the point that I'm embarrassed I asked the question in the first place!
To explore/manipulate my phone's file system, all I had to do was connect it to my computer via USB, and choose "Disk drive" as the connection type.
Then the storage on my phone shows up as two additional volumes: one for
/sdcard and one for
/emmc, which I can manipulate like any other volume.
Unfortunately, the approach as I've described it gives access only to those areas of the phone's file system that a user can modify without root access.
This approach is good enough for my purposes, but if you want free rein over your phone's file system, and don't mind going over to the "dark side" to get these powers, see this XDA Forum Thread.