How can I recursively copy files by file extension, preserving directory structure?

You can use find and cpio to do this

cd /top/level/to/copy
find . -name '*.txt' | cpio -pdm /path/to/destdir

(-updm for overwrite destination content.)

cd /source/path
find -type f -name \*.txt -exec install -D {} /dest/path/{} \;

Another approach

find . -name '*.txt' -exec rsync -R {} path/to/dext \;