The Office 365 Support team has just released a guided mail flow troubleshooter here:

http://aka.ms/mailflow

If you have users complaining that mail delivery takes too long, or that mails are remaining in their outbox, or that recipients are not receiving e-mails at all; the guided walkthrough will step you through the troubleshooting steps required to diagnose and resolve the problem

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Access the troubleshooter here: http://aka.ms/mailflow

For your reference, the following diagram shows many of the components involved with mail delivery from sender to receiver, any of which could be interfering with your email arriving: