Some options for you
If you just want to
copy
the directories from one place to another then use cp
cp -rp dir1 dir2 dir3 ... dest
The
-rp
flags tell cp to copy recursively and keep permissions/timestamps on the copies.
If you want to
move
the directories, permanently to a new location the use mv
mv dir1 dir2 dir3 ... dest
If you just need to provide access from your destination directory to your data, then perhaps you can just set up soft links
ln -s dir1 dir2 dir3 ... dest
This is fast, as it does not actually copy data anywhere, and if you modify any data in dir1 (e.g.), then it is available instantly to anyone looking at it from dest.
If you want to copy your data to dest, and you need to update regularly, but only a subset of files will be changed, then consider using rsync
rsync -ar dir1 dir2 dir3 ... dest
After the initial copy, this will copy only those files that are different between the source and destination directories, which will dramatically cut down the data transfer on copies after the first one.
All these commands support a -v option to be verbose about what its doing, so you can see what the command is doing.