Are there any Protestant denominations or churches that reject penal substitution? I come from a background in Protestant/Evangelical churches. The trademark salvation message in those churches is that Christ bore the wrath of God for your sins to satisfy his justice that we have offended by our sin; it's basically a teaching of the penal substitution theory woven into a call to repentance. John Piper also seems to declare the theory as gospel in his books.

Since becoming a catechumen in the Eastern Orthodox church I have rejected this theory as an acceptable way to explain salvation. It is my opinion of most Protestant/Evangelical churches that they teach the theory as fact (perhaps less so in for high church Protestants).

Are there any Protestant denominations (or individual churches) that explicitly reject the penal substitution theory, and more generally the idea that Christ's death was needed to "pay" or satisfy God the Father?

For reference, the teaching of the Eastern Orthodox church sometimes uses the ransom theory of atonement to explain Christ's death being a ransom paid to death. However, that is just one teaching that can help a Christian who is meditating on the meaning of His death.
Not all Anglicans have held to Penal Substitution Theory.  For example, the nineteenth to early 20th century Anglican bishop Charles Ellicott did not.  See his commentary on Galatians 3, for example.  The Thirty-Nine Articles of Religion does not express Penal Substitutionary Atonement.  So, if you look at some of the continuing Anglican denominations you may find a church that is Protestant but does not hold to Penal Substitution Theory. Some of those continuing denominations are "Anglo-Catholic", so I'm not sure they are actually Protestant. 

I am in a denomination called the Federation of Reformed Churches.  I do not hold to Penal Substitution Theory.  I've expressed this in a paper sent to my fellow Presbyters.  None of them attempted to bring any charge against me.  So, there are Reformed Presbyters who can tell the difference between theory and scriptural data and have some appreciation of the history of the doctrine of Christ as our propitiatory sacrifice.

It is interesting to note that Calvin did not write his view on penal substitution theory into either the Geneva Confession of 1536 or the later French Confession.  (There is some question as to whether Calvin wrote those documents or just influenced them.)  Calvin seems to have been more careful of such things than the Lutherans and many of the other Reformed, such as Ursinus.

There have been Reformed believers and are at least some now who do not hold to Penal Substitution Theory. Furthermore, it is not hard to find Arminian teachers who do hold to it.  So I do not think that holding to or not holding to the theory has any direct relationship to one's doctrine of free will.