Yes, the Medieval Church Evangelized, and yes, it Evangelized Muslims as well

 So I thought I would do a post on the Medieval Church's evangelism of the world in which it was passing through. It is commonly claimed, though false so, that Christians went to war with the Muslims without ever trying to convert them. That I will refute in this post.
 One of the most obvious examples of evangelism was by the Monks of Rome and Constantinople that brought Christianization to Europe by their teaching of the true Catholic faith during the early Middle Ages.  Irish Monks played an especially important role in the evangelism of Northern Europe.
 From St. Patrick going to Ireland in the 6th century, to St. Augustine of Canterbury going to England in the 7th, Monks not only Christianized, but frequently brought literacy and learning to the peoples of Europe.

Open air preaching was quite common throughout the Middle Ages. Friars would often go from town to town and preach people of Christ. This was especially common during Crusades. 

 John of Montecorvino was an Italian Franciscan that did missions in both China and India during the 13th century. 

 Saint Aurelius and Saint Natalia were Spanish Christians. This husband and wife were quite open and bold in public for their commitment to Christ.  Aurelius was born from a Muslim father and Christian mother, he and his wife Natallia were martyred in A. D. 852 for their confession of Christ. A monk by the name of George with them had spoken out publicly against the Prophet Mohammed.
 The above were not the only instances of Christians that tried to preach the gospel to the infidels as well as to other Christians.
 Open air preaching was quite common throughout the Middle Ages. Friars would often go from town to town and preach people of Christ. This was especially common during Crusades. 
 Pope Urban II called for the First Crusade in July of 1095 at Clermont France by an open air sermon. 
 Gerald of Wales went around Britannia preaching for the Third Crusade. 
 William of Rubrick was a Franciscian Missionary sent out by Saint Louis IX, King of France, during the Seventh Crusade to those in Eastern Europe and beyond. 
 There were several Jesuits that open air preached during the Reformation Erra. 
 The above are only a few examples. 

Saint Francis of Assisi preaching to the Muslims

Francis of Assisi believed even birds should here of the proclamation of the gospel 

A good book pictured above about Saint Francis preaching Christianity to the Muslims

The great Thomas Aquinas whom did have some dialogue with the Muslims pointed out their errors in the above book. 

Some sources for Further Information: The Medieval World by Anita Baker, Positively Medieval: The Surprising Dynamic Heroic Church of the Middle Ages by Jamie Blosser, Aquinas on Reasons for our Faith against the Muslims translated by Father Damian Fehlner,   Saint Francis of Assisi and the Conversion of the Muslims by Frank W. Rega, A Mirror of Medieval Wales: Gerald of Wales and his Journey of 1188 by Charles Kightly, The Mission of Friar William of Rubrick: his journey to the court of the Great Khan Mongkee, 1253-1255, Usborne World History: Medieval World by Jane Bingham,