Using "avprobe" I saw my MPEG4 movie has two audio streams. The Roku is finding the first, 5-speaker stream and playing it, instead of my desired stereo output.
So, the proper incantation to have normal audio is to tell avconv to use the first video stream, and skip the first audio stream:
avconv -i input.mp4 -codec copy -map 0 -map -0:a:0 output.mp4
Success! Now about those subtitles...