Billy, by Ohio law, you have to have a front and back plate. If your car doesn't have a holder, you have to make one. Check with your auto dealer about appropriate placement; I'm sure they've run into this problem many times with brand-new autos, so they should have the optimal solution for appearance's sake.
I don't know if the placing of the plate on the dashboard is legally compliant. At this particular point, we'd need to see the text of the law on plate display. Maybe you'd get help on finding such a text at one of the local licensing offices (Miracle Mile, Heatherdowns, downtown).
Whew! I did some searching here, and found it:
4503.21 Display of license plates and validation stickers or temporary license placard or windshield sticker.
(A) No person who is the owner or operator of a motor vehicle shall fail to display in plain view on the front and rear of the motor vehicle the distinctive number and registration mark, including any county identification sticker and any validation sticker issued under sections 4503.19 and 4503.191 of the Revised Code, furnished by the director of public safety, except that a manufacturer of motor vehicles or dealer therein, the holder of an in transit permit, and the owner or operator of a motorcycle, motorized bicycle, manufactured home, mobile home, trailer, or semitrailer shall display on the rear only. A motor vehicle that is issued two license plates shall display the validation sticker only on the rear license plate, except that a commercial tractor that does not receive an apportioned license plate under the international registration plan shall display the validation sticker on the front of the commercial tractor. An apportioned vehicle receiving an apportioned license plate under the international registration plan shall display the license plate only on the front of a commercial tractor and on the rear of all other vehicles. All license plates shall be securely fastened so as not to swing, and shall not be covered by any material that obstructs their visibility.
No person to whom a temporary license placard or windshield sticker has been issued for the use of a motor vehicle under section 4503.182 of the Revised Code, and no operator of that motor vehicle, shall fail to display the temporary license placard in plain view from the rear of the vehicle either in the rear window or on an external rear surface of the motor vehicle, or fail to display the windshield sticker in plain view on the rear window of the motor vehicle. No temporary license placard or windshield sticker shall be covered by any material that obstructs its visibility.
(B) Whoever violates this section is guilty of a minor misdemeanor.
Effective Date: 01-01-2004
It seems that the specifications of "front and rear of the motor vehicle" and "shall be securely fastened" and "shall not be covered by any material that obstructs their visibility" would deny you the right to display the plate on your dashboard.
In addition, the law mentions windows placement only for the "temporary license placard or windshield sticker", so that strongly supports the idea that the license plate is only mounted on the exterior front and back of the vehicle.