If your OM-D has a Manual Flash mode, it could work with your SB-600 if you add a Nikon SU-4 optical slave. You would be limited to Manual flash mode and for the most consistant results need to have the camera in Manual exposure mode as well.
You could of course do as Marty suggests. But the SB600 has to operate in manual mode and you'll need to calculate the exposure using the guide number of the output strength you choose and the distance to the subject to attain a correct f stop. I always find that cumbersome since the SB600 does not have dial display to calculate it for you.
So how about using a flash, on camera, in Auto mode that does the power and distance calculation for you and tells you what f stop to use depending on the ISO being used. For my OM-D I use the Nikon SB-15 which has two Auto modes and is extremely easy to use. I just put my OM-D in M mod, set the shutter speed to 1/125 and set the aperture to what the flash calculator tells me. I get perfect exposures every time. The good news is you can get a nice used SB-15 for about $15. Of course there are many speedlights that can do the same thing. I just find the SB-15 easy to use and a convenient size.
I have a set of Cactus V5 radio triggers that work with the OMD and the sb600. You're stuck with pure manual settings but it does work. And it's probably a bit more reliable than an optical trigger since you're not limited to line of sight.