To expand a bit on what Adam just said, DoF is determined by Focal Length, aperture, and distance from camera to subject. Change any of those three things and your DoF changes.
A lot of sites don't always explicitly state WHY the DoF changes when you go to a micro-4/3rds sensor camera from a full frame camera or an APS sensor camera. It isn't the sensor size at all. Instead because the crop factor changes your field of view changes. Therefore to get the same magnification (how your subject fills the frame) when you switch crop factor is that you have to change either the focal length being used or the distance to your subject. THOSE are actually what change the DoF, not the sensor.
If you took an Olympus OM film camera with an old OM lens mounted on it, put it on a tripod, took your photo, then removed the OM camera and instead placing an OM-D camera behind that same lens not changing the distance (tripod stays put) nor changing the aperture of that lens, the resulting photo would have identical DoF as the film camera. What would look different about the shots is the OM-D version would have a much smaller field of views (if your film shot was a head & shoulders shot, your OM-D shot would now be a lips & nose shot

). In the real world, with the OM-D you would need to move the tripod back further to frame the shot the same or leave the tripod put and switch to a wider focal length.