Mailinator automatic checker for new e-mails
Overall it looks great, but here are a couple of thoughts:


print('New email received.'): maybe print a snippet form the message or just the time stamp so you have some idea on when the new email arrived (instead of just having a stream of same strings stating "New email received.").
I'd consider moving the inbox operations into its own class. Its __init__ could be __init__(inbox, api): .... CHECKING_FREQ could be a keyword argument to this, defaulting to 30, but configurable through the input params when running the script. The class could have a fetch_inbox (merging your get_inbox and check_inbox). And finally the has_new_items (just rename your is_new method).
Your wait_email is what I'd call check_inbox because for checking the inbox you first have to fetch the inbox and then check the message timestamps on what was returned.
I feel someone might have a good suggestion on how to do that infinite loop, but I can't think of a good way right now. All I know is it feels a little odd (or maybe it's fine :).


Hopefully this helps.