How can I copy a directory structure but ignore symlinks?

Moved from question into answer:

As h3rrmiller pointed out, I was able to achieve this with rsync by using the --no-links option.

According to the documentation of the cp command, you can use -d option:

‘-d’

Copy symbolic links as symbolic links rather than copying the files that they point to, and preserve hard links between source files in the copies. Equivalent to --no-dereference --preserve=links.