Check these things:
If you are lucky, it might be one of the tires.
Check visual condition of tires. Any humps or lumps on surface of tire you can see? If so, replace or install spare and retest at speed. Have garage or tire shop check for proper balance, broken cord in tire, tire wear, etc...
However, it will probably be worn suspension parts. This is harder to diagnose yourself and should be diagnosed by a certified mechanic who knows what to look for. Ask around with friends or co-workers for a reputable shop. (They do exist, although rare.)
Common problems on early Odysseys are: worn out sway-bar bushings and end-links. Less common but still prevalent are worn tie-rod ends.
Struts and associated hardware need to be replaced every 75k miles depending on service conditions. Occasionally (but rarely) Honda CV-joints will become defective--usually due to cracks or splits in rubber seals.
If you are handy you can look at the rubber bushings connecting metal parts to each other and the frame on the front-end. If these look cracked, swelled or worn they should be replaced and an alignment check performed by qualified mechanic.
Wheels are out of balance
shifted or broken belt in the tires
More than likely a tire out of balance. Have the tires balanced and inspected for a defect.
balance front tires
Get you front end aligned, that should take care of the shakes.
Sounds like wheel balance, but check the steering dampner.
try having the wheels balanced. It could also be un-even wear in the tyres especially the front ones.
Check into a new steering dampner, it's a common cause for shake at speed in a Jeep, but it could also be alignment or balance.
have your front tires rebalanced.
More than likely you have a tire out of balance. Have your tires rotated, inspected, and balanced.
Tune up
I needed new brakes when my 2003 Vue shook at 50mph. Once I had the back brakes changed the shaking stopped.