If you’re looking for a simple solution, with a free option, Zoho may be something you’d be interested in. All that is required for the initial setup is setting your domains MX records to theirs. They’ve been a great alternative to Google Apps, and they still offer a free option (of course, it’s limited - you’re not going to be allowed to create unlimited accounts or anything - for that, you’d need to focus on your own setup or pay)
Alternative, if you have programming experience in PHP, Python or Ruby (just a few examples), you can always use services such as MailGun, Mandrill, SendGrid etc. You’d simply setup your MX to match theirs and use their CP to add “accounts” (which function like virtual e-mail accounts). When one of these accounts receives an e-mail, a webhook would be fired off and from there, the webhook would route the e-mail to an application URI (i.e. something you’ve self-coded) and it would then be up to the application to handle the incoming POST request (which would include all the mail headers, the body, etc). It’s not something everyone does, though it is an option and it does relieve the burden of having to manage your own mail server. Instead, you manage your codebase, which for many, is much easier.