I am looking for sites from which to download classical Chinese music.
 The sites must predominately have complete compositions (i.e., not
samples or excerpts), not require registration fees, and provide the
music in MP3 format.

By classical Chinese Music, do you mean Cantonese opera?

Any music prior to 1900.

try these links to see if these are kind of the songs you are looking for:

http://www.chinakongzi.net/2550/music/download/music1.htm

http://www.goyoyo.com.cn/netter/mp34.shtml

http://nlnlrr.myetang.com/sdmq.htm