Adding a particle (準体助詞) "から" to the verb 調べて (like...きちんと調べて"から"モノを言いなさい!) would make it simpler I guess. It'd literally be like; きちんと=in a proper way/thoroughly 調べてから=after you investigate/after you do a research 物を=a thing/things/something/anything, etc. 言いなさい=say (imperative) I think the whole sentence would be along the lines of; "You should say what you say after you've done a thorough research" (which I think carries a fairly criticizing tone.)

This means, people should research or know about what they are going to say to someone before doing it. Or you also may be said that if what you are saying is wrong and as if you say it as correct. However, the later example would probably fit your question.