Preserve directory tree while copying files with cp
With rsync:
find ./ -name myFile.txt -print0|rsync -0adv --files-from=- ./ ../newTree/
Without rsync:
You can find all files, for each file you create the directory in the newTree, and copy the file to it.
for file in */myFile.txt; do dir=$(dirname "$file") mkdir -p "../newTree/$dir" cp "$file" "../newTree/$dir"done