I think you cannot have a medieval game without attention to religion, and most particularly to christianity. However, as a medievalist, I am constantly dissapointed by how simplistic religion is depicted in popular media.
My personal expertise rests mostly with western Europe and to a certain extent with the Baltic Crusades. However some points may be worthwhile exploring for Bohemian history as well.
First of all, when we speak about christianity, we are not just speaking about theology, but about a collectively shared culture. It was a framework of meaning commonly shared by everyone, but often in different ways.
On the lowest level, the parish was often the main organizational level. From the Gregorian Reforms onwards (11th century) the Church, but also many noble rulers, actively stimulated the deepening of the faith, establishing many parish churches. In most parts of Europe, most nucleated villages had a parish church, which was often at the center not just of the religious life, but of many cultural and social interactions. For instance, important deals were often made at the portal of the church or even in the graveyard surrounding the church. Theatre, often biblical plays, took place in front of the church, etc. Particularly the observance of the sacraments (which, for most part, are also important social rituals) was held very important (many attacks against heresy focus on the observance of the sacraments).
However, as far as deep theology or strict observance and orthodoxy went, medieval religion was rather vague and highly variable. Many ‘popular practices’ persisted throughout the Middle Ages, and parish priests were seldomly highly educated people. THe priests could read (they mainly received a trivium education, but most of them had not attained university or even a decent cathedral school and were thus poorly instructed in the deeper theology of scripture).
Religion was therefore mostly based on the typical stories and often instructed in a very visceral and iconographic fashion (sculptures, glass windows, but also illuminations in bible books or psalters played an important role). The stress, of course, laid on the gospels with some typical prefigurations taken from the Old Testament. The moral codes were greatly drawn from the epistles, certainly that from Paul (for instance, on marriage, the responsabilities of men and women, etc.). And, to be sure, great stress was also laid on the apocalyptic stories (even more after the the Plague (1348-1353): something to look out for in the afterlife or something to fear (whatever way you take it).
Parish priests were often local boys, who in one way or another had learn to read (sometimes even from the former priest) and officiated by the bishop. Although priests had to be celibatary in theory, most of them lived together and could even be married (I graded a master’s dissertation a few years ago that actually showed that this was more rule than exception, it was only with the Tridentine Reforms in the 16th century that this changed).
The parish priest was therefore often a kind of educated member of the community, often also taking the role of the representative of the community when dealing with ‘higher authorities’. ON the other hand, he was an influential figure, often belonging to the more important peasant families.
Of course, the Church was bigger than its parish priest. IN fact, more important than the pope (with papal authority being at an all time low in the early fifteenth century), was episcopal authority. Now bishops are something else and should be considered not just religious but also very important secular rulers. Bishops often held vast amounts of land and in the German Empire they could even have there own principalities. Cologne, Trier or Liege are just a few examples of these positions. Consequently, the bishop’s cathedra was an important position of power for many noble families and often the subject of much dispute. Every noble family had one of its sons send of to the universities of Europe to study theology in the hope to later place him on one of these prestigious seats. The families that did not make it (or more regional noble families) would have theirs sons take up position in the cathedral as a canon (sometimes explicitely to oppose the rule of the bishop, you often see this when an outsider is nominated bishop in a particular region). Although bishops often stressed on theology (and they had the knowledge of course), many of these claims were rarely simply religious, but had to do with a power display against other factions or groups within society as a whole or the nobility in particular.
And then there are the monks: Again a very important group in medieval history. Not in the least because monasteries often held vast lands, but better than noble lords or bishops, were often much better equipped to deal with them. Monasteries carefully managed there lands, often holding a close eye on the income rendered from the different villages belonging to their estates. Such monasteries therefore often housed many second or third sons of local or regional noble houses.
Medieval monasticism, however, is not as straightforward as it may appear. There are many different ‘orders’. The benedictines (although technically not really an order, as they are not centrally governed) were the oldest, and therefore often holding great monesteries with vast amounts of lands tied to them. Because of this long tradition of wealth gathering, benedictine monasteries were often considered luxurious places to live in. More than often, when kings or notable princes visited a particular location, they opted to stay in a monastery rather than in the damp, moisty keeps not solely for pious reasons. Next to that, there were the Cistercians. An order known for its more rigorous interpretation of the rule of Saint Benedict (although also that had weakened in many monasteries by the beginning of the 15th century) and excellent planners (if I am not mistaken, a lot of Cisterian foundations were also made in Eastern Europe, not in the least for their planning skills). Another typical characteristic is the Maria devotion of the Cisteriancs, which gained great support amongst the general population as a more ‘emotional’ and ‘mystically’ driven cult than the saintly cults of their benedictine counterparts. A last important note on the Cistercians is that from the 13th century onwards, a lot of women’s convents were founded in the Cistercian Order (although they were strongly superivzed by the male members, of course).
A last group of orders were the mendicant orders, with the Fransciscans and Dominicans as most important orders. These orders took a completely different strategy than the others, since they mainly established themselves in urban centers. Fransciscans are particularly known for their vow of poverty and their convents would rarely accept to much material wealth (so no huge estates as their Benedictine counterparts). Dominicans were known for their preaching and therefore there learning (it is for this stress on learning that the Dominicans were put in charge of the Inquisition, which certainly did not always involve gruesome torture, but often entailed rather fair and modern juridical procedures, at least untill one of them devised the ‘malleus malificorum’, but that is later in the 15th century).
Nonetheless, you would find these dominicans also often outdoors, certainly on pilgrim roads. They were notorious preachers and not so much by preaching high ecclesiastic retorics, but rather by telling stories that many people could understand. The dominicans compiled small stories of saint lives, the gospels, or moralistic stories in books called ‘legendaria’ (the legendarium aureum being a very noticeable example). Although these stories were compiled in Latin, these preachers had the ability to instantly translate these stories and deliver them to general audiences in an appealing theatrical way (in the process inspiring a more moral way of living of course). That these stories must have held some influence is even noticeable today: Red Riding Hood is derived from one of these stories which needed to warn young girls for the many ‘wolves’, as to chastely preserve them for the sanctity of marriage.
Now with the preachers, we have come to the more greyish figures of religion. In the Middle Ages, popular preachers or ultra pious men were bountifull. Historians often speak about a sort of clerical proletariat which held popular preachings often in exchange of money or support. Some of these preachers would grow into noticeable heretics (like Waldo or Huss), where others were sanctified (such as Francis of Assisi), and most were not noted by the historical sources. Next to these preachers, also heremitism ocured. People who drew away from public life and became ascetic wanderers, often exiling themselves in barren natural conditions.
Now this is just a small overview of how religion in the Middle Ages worked, but I think there are many elements that could be developped in game (small plots, interesting characters or just simply as a silent background with a lot of history that will add to the feel of authenticity)