MySQL Export of MD5 password fields to new MySQL server
First step would be to verify that the MD5 hashes in your new database match the ones from the old database. If they do not, then there was an error with your export/import, perhaps you did an MD5 of an MD5 or something. If the hashes match perfectly, then it sounds like your php code that handles logins is causing the problem.
We really need a lot more information. Are you salting the passwords at all? How does your login process work? Like tfinch2 said, you should try and keep the versions of your server software the same to try and rule those out.
Consequences will never be the same!