The Odyssey, like all large vans, is having transmission problems. Transmission, suspension and brakes were under-designed. Problems may have been solved in new models (2004+).
We blew through our first tranny at 77,000 km (47k miles), and our second now at 92,000(57k miles). Fool me once, shame on you - fool me twice, shame on me!
Honda has been on a re-manufactured parts program for more than 10 years. Our service manager advises that about 6 months ago Honda announced major improvements to the re-manufacturing process: better designed parts, more stringent pass/fail criteria, etc. Problems with replacement transmissions have been common, but they are confident (FWIW!) they have solved the problems.
New models have a completely redesigned 5-speed auto tranny.
Persons who've had chronic transmission failure should call or write their Honda national customer service department, so a file can be opened on your vehicle, in case there's an issue with out-of-warranty failures.
Brakes have also been a big problem on this vehicle. The rears were drums, not big enough to handle the weight of a full passenger load in hilly terrain, and our drums (like many others) overheated continually and eventually warped. They were re-machined under warranty. New models have rear discs.