The simple answer to this question would be "yes", however this is only half the truth.
Quite a few Linux distributions do not include MP3 playback software due to the patent situation of the MP3 format. These patent issues do not affect the user directly, so most distributions make it fairly easy to install MP3 support over the Internet after the installation, making it possible to legally play MP3 files, for non-commercial use.
DVDs, on the other hand, are a more difficult situation. Most, if not all, DVDs are encrypted using a system called "CSS". There is a widely supported free software, called
libdvdcss, capable of decrypting DVD videos. However, circumventing the restriction on your DVDs is illegal in many parts of the world, including the USA and the European Union.
There is a legal solution to play your DVDs on Linux where
libdvdcss is illegal. The Fluendo company has developed an (non-free) application for that purpose which can be purchased from their website.
Non-encrypted DVDs, which include most home-made DVDs, play perfectly well with only free software.