To achieve what you're looking for, without using a service like Dyn, you'd use Access Keys to access your server
1. Find out details on, or create your Access Key via
Options > Media Network > Access Key. If you click on the button to the left of your current Access Key, it gives a few handy options for generating an invite like this one:
Hi,
If you have a copy of Media Center on your PC, you can play media from my library by selecting 'File > Library > Connect to Library...' and then entering the following access key:
<ACCESS KEY>
You can also use a web browser on a PC or a portable device as a remote control for my library by using this link:
http://webremote.jriver.com/<ACCESS KEY>
You can also play media from my library on your iPhone or other device by using this link (on your device):
http://webplay.jriver.com/<ACCESS KEY>
Thanks. And don't laugh at me for having Abba in my library.
2. Configure client copies of MC, or apps like Gizmo and JRemote, to connect to your server via the Access Key instead of using an IP address. Use the URL from the greeting message to make your bookmarks.
Bear in mind, this does require an active Internet connection (at both server and client ends) to function (whereas using an internal LAN IP will work at your house even if your Internet connection is down). Of course, you can always add more than one "copy" of the network Library to MC from
within the Library Manager, including a direct LAN IP version should one ever be needed, so that's not a big concern.
When accessed via an Access Key, MC connects through JRiver's servers, which can resolve your server's public and private addresses even if they change. This works just like many "GoToMyPC" type services operate, including Back to My Mac from Apple.
3. Forward the proper TCP port through the firewall on your LAN's Gateway/Router to the server inside your LAN. The specific instructions for how to do this, and the requirements, depend on your specific gateway device. With most consumer routers, this will require you to set whatever machine acts as your server to use a static IP address inside your LAN (but this has no bearing on your dynamic Internet address, only the private 192.168.x.x or 10.x.x.x address used inside your private network). Then, you tell the router to forward incoming traffic on that particular port to the machine at address 192.168.whatever. You typically do this by accessing a web "utility page" that the router creates inside your network, though it could be accessed via (as with Apple's AirPort devices) an actual application you run on your computer.
The Port MC is configured to use is specified at:
Options > Media Network > Advanced > TCP Port. I believe 52199 is the default, but I changed mine so long ago that I don't remember, and I don't see it
explicitly stated as the default anywhere in the wiki (though that's the port number used commonly in all the examples, so I really think it is).
You can probably look up instructions specific to your router on the website for the manufacturer of the device. Look for FAQs about Port Forwarding. There are also some general-purpose Port Forwarding websites out there with instructions compiled for a variety of devices. However, be wary of some of the scammier varieties that try to sell instructions, or "wizards" to do it for you (which are often little more than how to videos wrapped in an application).
I should add... While you are at it,
update the firmware on your device. Infected and compromised gateway devices are an epidemic. If possible, you should consider replacing the stock firmware on your router (assuming you own it and aren't renting it from your ISP) with a community-powered one like
dd-wrt.
4. If you have the OSX Firewall enabled, ensure that it is allowing incoming traffic on the specified Port. On Mavericks you do this by going to:
System Preferences > Security & Privacy > Firewall > Firewall Options
If
Block all incoming connections is
unchecked, as it is by default, then you'll probably be okay without further changes. However, if you want to be sure or you experience difficulty, you can add a application-specific exception by clicking the plus sign and choosing MC.
If this still doesn't work, after completing the above, see if you can solve it by disabling the OSX Firewall. If so, you can open the port manually at the terminal. Post back for further help in the unlikely event that you need to do this.