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Open OmniOutliner. Open the OmniOutliner Preferences. Click on the General tab, if not already selected. Click the Edit New Document Template button. In the resulting template window, click on the Utilities button to open the drawer, then select Whole Document at the bottom left. Bring up the Document inspector (command-4), click on the Display tab (looks like a display with a checkbox in it), uncheck "Show status checkboxes". File->Save and you should be all done.

I generated these instructions from OmniOutliner Pro if you've got OmniOutliner and one or more of the details is different, hopefully you can connect the dots.