Sounds like you have looked at the obvious stuff.
I would methodically start at the back end of the bike and check every thing for proper adjustment, damage ect. including bearings, spoke tension. Don't overlook a single inch. I would pay extra attention to frt. and rear wheel alignment and headset and obviously check for frame cracks/ damage paying special attention around the bottom bracket, drop outs and head tube. Make sure the head tube is not a lose fit for the headset. Even a slight lose fit due to a crack that allows play here could cause this problem and portend a sudden and serious failure.
Another thing that can cause this is wheel reflectors (I hate HATE HATE them) and I have even seen tubes with waaaay too many patches and/or big chunky old school patches that do not have feathered edges causing the shakes. Also, try removing the tires and rotating them 90 degrees in their relationship to the rims.
Personally, I would not ride the bike at higher speeds until I had ascertained the cause of this problem.