This is going to be a really quick one, because I'm mostly posting this so that the next time I google it, I get my own blog instead of some StackOverflow posts and what not.
With the Exchange Online PowerShell V2 Module, you can use your Partner Delegation privileges to connect to your customer's Exchange Online tenants:
Connect-ExchangeOnline -UserPrincipalName <YOUR PARTNER UPN> -DelegatedOrganization <YOUR CUSTOMERS PRIMARY DOMAIN>
The part that always trips me up, is that I assume the
-DelegatedOrganization arguments is looking for the tenant, but it's not. It's looking for the primary domain. So for example, the tenant contoso.onmicrosoft.com is not the correct value; contoso.com is:
Connect-ExchangeOnline -UserPrincipalName email@example.com -DelegatedOrganization contoso.com