It depends on who you ask:
According to the owner's manual, it cannot be dinghy towed (4 wheel on the ground flat-towed). The manual states:
"Your vehicle was not designed to be towed behind a motorhome".
However, for the 2008 Odyssey, Remco (http://www.remcoindustries.com/Towing/
"Officially from both Honda and REMCO the vehicle is not towable. However, we at Remco have a large number of customers who are flat-towing that vehicle behind their Motor Home and they are not having any issues. Follow the "Emergency Towing" procedures in the owner's manual to a "T" except ignore the restrictions and follow those procedures each and every time you tow. NOTE: If you choose tow your Honda, and have any damage to the Transmission as a result, you will have to pay for that expense out of pocket. It is nothing that Honda or Remco will warranty. Also, it should be noted that Remco sells no aftermarket product (Lube Pump or otherwise) to make this car towable."
Sounds like you are between yes and no. However, since a 2008 will be fully depreciated, if you damage it, you won't be out as much as if you damaged a new car.
We ran into the same situation with our 2009 Nissan Maxima; with it's CVT (continuously variable transmission) it could not be towed with 4 wheels on the ground. At first, I replaced it with a 2014 Ford Taurus which can be dinghy towed, but I could not bring myself to cut up the front fascia of a new car. So I ended up buying a second car; a 2002 Pontiac Grand Am, can be also towed with 4 wheels on the ground.
Most owner's manuals have a "Recreational Towing" guide that will indicate whether or not you can tow it behind a motorhome.
Good luck on your adventures.