The answer depends on how you define "simultaneously," but in most cases, such a maneuver is probably legal.
Ohio Revised Code section 4511.39 requires you to signal a lane change "continuously during not less than the last one hundred feet traveled ... before turning."
I think the most natural reading would be that you need to signal your intent to move from Lane 1 to Lane 2 for at least 100 feet, and then signal your intent to move from Lane 2 to Lane 3 for at least 100 feet.
Note, though, that at some speeds, the 100-foot requirement is practically not a requirement at all. If you're traveling at 70 miles per hour, which is Ohio's default freeway speed limit, you'll cover 100 feet in less than one second. So although you probably couldn't just do a hard jerk from Lane 1 to Lane 3, you should be fine if your car spends one full second entirely in Lane 2.
FWIW: I don't believe that the Florida trooper has accurately characterized the law, either. Under F.S. 316.085, the only requirement before changing a lane is that the driver ensure:
that the vehicle is not being approached or passed by any other vehicle in the lane or on the side to which the driver desires to move and that the move can be completely made with safety and without interfering with the safe operation of any vehicle approaching from the same direction.
F.S. 316.155 requires the driver to use a turn signal for at least 100 feet before the lane change, but it doesn't require that he move "slowly" from one lane to the next, nor does it prohibit "crossing multiple lanes in one movement." It merely requires the driver to ensure the lane change "can be made with reasonable safety."