I thought the phrase was “Speak your peace” but my roommate says it’s “Speak your piece.” What’s correct?
It’s “Speak your piece”. As in a “piece of your mind”.
Judging by google searches, Speak your peace is preferred to Speak your piece, by a margin of 6,270 to 2,220.
From bartleby.com:
the tyranny of democracy
A Google search confirms: Lots of people are dumb.
Google also finds nearly 2.5 million hits for the non-existent word, “alot”. Even Google suggests helpfully, “Did you mean a lot?”
Ah, perhaps the simplicity of counting Google hits. 
If you take a look at the individual hits, you will find that they don’t really reflect “speak your peace”. Here is a typical example:
"Speak your Piece for Peace" gathering on Feb…
It turns out that a lot of peace activist groups have adopted the “speak your peace” phrase as encouragement to “witness” your dedication to the peace movement.
There are a couple of other interesting types of hits. There are the religious ones. A lot of old prayers are of the form “Oh Lord, let me speak of your peace…”
It also appears that some people have converted “Speak now or forever hold your peace” to “Speak your peace”.
Then, of course, there are some that are just plain dumb.
English is a fascinating language.