I have seen people ask about this very same problem and now that I have it fixed I'd like to share the steps taken to resolve it. Kudos to Apple's Product Specialists in spending a lot of time troubleshooting to find the root cause then to fix it.
My setup:
I have a Powerbook and Macbook Pro, both set up to sync everything with MobileMe. The problem started when I added the Macbook Pro to sync. From then on every time I would launch Mail.app it would ask for the password for two of my non-MobileMe email accounts (IMAP). I would enter their passwords and check Remember password in my Keychain, but next time Mail.app would launch it would ask again. If I would open Mail > Preferences and look at the username and passwords fields for the accounts, the password field would be empty. Again, this was just for the Macbook Pro. The Powerbook never had a problem.
Below are the steps taken to fix the issue. If you attempt these steps you do so at your own risk.
Problem: Mail does not retain non-Mobile email account passwords when syncing between computers
Solution:
1) On the affected computer, open MobileMe System Preferences and disable Syncing.
2) Move the folder ~Library/Keychains to Desktop
3) In Keychain Access -> Preferences choose Reset My Keychain
4) Reboot computer
5) Open MobileMe System Preferences and log into MobileMe
6) Under the Sync pane, check Synchronize with MobileMe and select all services
7) Click Advanced... then Reset Sync Data...
8) In the drop down select Keychains and then choose to pull data from MobileMe to the computer and click Replace
9) Launch Mail.app
10) If prompted to enter passwords for the mail accounts, exit Mail.app
11) Open Keychain Access and remove all entries (IMAP and SMTP) for affected email accounts
12) Launch Mail.app
13) When prompted to enter passwords for the accounts, enter password and check Remember password in my Keychain
14) Verify password took by opening Mail -> Preferences, select the affected account and verify password field is not blank
15) If passwords were retained, open MobileMe System Preferences
16) Under the Sync pane, click Advanced... then Reset Sync Data...
17) In the drop down choose Keychains and then choose to push data from the computer to MobileMe and click Replace
18) On other computers being synched quit Mail.app
19) Force a sync in MobileMe System Preferences
20) Relaunch Mail.app and verify that you are not asked for a password
21) If asked for a password, quit Mail.app and perform steps 1 through 9 on each computer being synched
22) After these steps have been completed force a full sync on each computer one at a time to verify the issue has been resolved
My setup:
I have a Powerbook and Macbook Pro, both set up to sync everything with MobileMe. The problem started when I added the Macbook Pro to sync. From then on every time I would launch Mail.app it would ask for the password for two of my non-MobileMe email accounts (IMAP). I would enter their passwords and check Remember password in my Keychain, but next time Mail.app would launch it would ask again. If I would open Mail > Preferences and look at the username and passwords fields for the accounts, the password field would be empty. Again, this was just for the Macbook Pro. The Powerbook never had a problem.
Below are the steps taken to fix the issue. If you attempt these steps you do so at your own risk.
Problem: Mail does not retain non-Mobile email account passwords when syncing between computers
Solution:
1) On the affected computer, open MobileMe System Preferences and disable Syncing.
2) Move the folder ~Library/Keychains to Desktop
3) In Keychain Access -> Preferences choose Reset My Keychain
4) Reboot computer
5) Open MobileMe System Preferences and log into MobileMe
6) Under the Sync pane, check Synchronize with MobileMe and select all services
7) Click Advanced... then Reset Sync Data...
8) In the drop down select Keychains and then choose to pull data from MobileMe to the computer and click Replace
9) Launch Mail.app
10) If prompted to enter passwords for the mail accounts, exit Mail.app
11) Open Keychain Access and remove all entries (IMAP and SMTP) for affected email accounts
12) Launch Mail.app
13) When prompted to enter passwords for the accounts, enter password and check Remember password in my Keychain
14) Verify password took by opening Mail -> Preferences, select the affected account and verify password field is not blank
15) If passwords were retained, open MobileMe System Preferences
16) Under the Sync pane, click Advanced... then Reset Sync Data...
17) In the drop down choose Keychains and then choose to push data from the computer to MobileMe and click Replace
18) On other computers being synched quit Mail.app
19) Force a sync in MobileMe System Preferences
20) Relaunch Mail.app and verify that you are not asked for a password
21) If asked for a password, quit Mail.app and perform steps 1 through 9 on each computer being synched
22) After these steps have been completed force a full sync on each computer one at a time to verify the issue has been resolved