We are migrating from a dedicated server with a local virtual SMTP server to Windows Azure. As far as I can tell Windows Azure does not allow a local SMTP server to be setup in IIS. The SMTP Server option in the management console is missing. How would I setup an email relay so that I can have a .net web application send emails from a Windows Server 2012 virtual machine in Azure?
This is more like corollary to what @mcollier has stated. Given that you are already on Azure and you get 25,000 free emails (there are higher plans as well) with your subscription as well.
You can configure a Virtual SMTP server which relays to the sendgrid services. For the development perspective you will have the view of using your own SMTP server / service. Setting up a sendgrid based service is explained in this link.
I have used sendgrid earlier and my experience with this was amazing. As @mcollier has pointed out, using services like Sendgrid will give good chance of staying away from spam problem, if you try to setup and use the SMTP service of your own because of the reverse lookup etc.
PS: The above Virtual SMTP Server can be replaced with Amazon SES, which is equivalently a good service. Check out more info here.