You can clear all logs in the GUI in one step. While watching one logfile in check_mk you can choose the option Logs - All Logfiles and then click on “Clear Logs”. Deleting them under ~var/check_mk/logwatch may result to delete logfile informations without viewing them before, but is is a good solution for clearing all stored logfiles and getting the checks green 
Greylog is a centralized solution for collecting, storing and analyzing logfiles. While check_m is not storing logfiles graylog can store them and is a powerful tool for working with logfiles in larger enviroments.