Notwithstanding the fact that the best way to move data forward to a new version is to do an in-place upgrade, if you're comfortable with doing backups of selective tables, you might choose to move the data manually between databases.
In which case, I suggest this approach:
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Upgrade the old database ( or a copy of it ) to the new version via zc_install/index.php Database Upgrade option
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Backup the appropriate tables from the now-upgraded database.
A reference list to identify "which" tables are desired is here:
http://www.zen-cart.com/forum/showthread.php?t=27393&highlight=selective+database+backup
(In phpMyAdmin, best to select "Complete Inserts" and "Extended Inserts" to keep the files smaller and more efficient in reloading. GZipping is nice too for faster download/upload.)
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Drop those same tables from your "new"/fresh database.
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Import the tables from your backup in step 2 above.