Catalysis of the reaction: DNA containing 3-methylbase + H2O = DNA with abasic site + 3-methylbase. This reaction is the hydrolysis of DNA by cleavage of the N-C1' glycosidic bond between the damaged DNA 3-methylpurine or 3-methylpyrimidine base and the deoxyribose sugar to remove the methylated base, leaving an apurinic or apyrimidinic site.