Each city should directly elect its own mayors. By letting citizens directly elect these individuals, a mayor is chosen by the people and must answer directly to them rather than corporations or outside organizations. With that in mind, direct election of mayors does occur in some jurisdictions with great results.
Modern voting machines makes it easy to elect local officials and creates more accountability in the electoral process. Mayors should be elected by their constituents, not by city councils, since the mayor is the entire city's voice on the governing body. Eventually, presidents should also be directly elected instead of using the outdated Electoral College method.