which was the common language spoken at that time?
Proto-Mayan.
Maya civilization
Before 2000 BC, the Maya spoke a single language, dubbed proto-Mayan by linguists. 
who held the power in the maya civilization The king. when did this civilization develop? 2600 BC. which materials and technology are used by the people of that time to build their cities? Neolithic technology; stone, limestone, volcanic tuff, sandstone and fired bricks.
wiki:15310968
Maya civilization [SEP] Language
Before 2000 BC, the Maya spoke a single language, dubbed proto-Mayan by linguists. Linguistic analysis of reconstructed Proto-Mayan vocabulary suggests that the original Proto-Mayan homeland was in the western or northern Guatemalan Highlands, although the evidence is not conclusive. Proto-Mayan diverged during the Preclassic period to form the major Mayan language groups that make up the family, including Huastecan, Greater Kʼicheʼan, Greater Qʼanjobalan, Mamean, Tzʼeltalan-Chʼolan, and Yucatecan. These groups diverged further during the pre-Columbian era to form over 30 languages that have survived into modern times. The language of almost all Classic Maya texts over the entire Maya area has been identified as Chʼolan; Late Preclassic text from Kaminaljuyu, in the highlands, also appears to be in, or related to, Chʼolan.