Final answer:
Byzantine cultural influence in Eastern Europe was primarily spread through religious missions, with missionaries like Saints Cyril and Methodius playing significant roles in converting Slavic peoples and promoting literacy with the Cyrillic script.
Step-by-step explanation:
Byzantine cultural influence was spread to Eastern European peoples primarily through religious missions. Constantinople, the capital of the Byzantine Empire, was instrumental in the preservation and spread of Christianity throughout Eastern Europe. The Byzantines saw themselves as the continuation of the Roman Empire and, through their missionary activities, played a key role in the religious and cultural development of the region. Orthodox Christian missionaries, such as Saints Cyril and Methodius, were active in converting Slavic peoples and are celebrated for creating the Cyrillic script, which helped spread literacy and liturgy in the vernacular.
While trade routes and marriage alliances also contributed to cultural exchange, it was the widespread efforts of Christianization, supported by the Byzantine religious and political establishment, that had the most enduring impact on the cultural landscape of Eastern Europe.