If you have an Android device you can probably use MHL (if you want to see things on the screen) at which point you can use MediaMonkey for Android (iOS devices also have HDMI converters, but not MediaMonkey). I also believe BubbleUPnP on Android can stream to Chromecast (don't have Chromecast), I've used to stream MMW to DLNA clients and it works well for me (I stream video though). If you have Android 4.2 or newer you likely can use Miracast screen mirroring as well.

Otherwise you'll need to find a different hardware solution. Be wary though as many streaming devices (Roku is an example) don't have DLNA capabilities or may lack format support. There are however a plethora of client options like game consoles, networked receivers and dedicated media streamers available if you don't have a smart TV.

I would like to stream music from MediaMonkey to my non-smart TV. What would be a good/cheap client device that would attach to my TV and allow this?

Right now I use Plex to stream from one of my computers to my Chromecast. I like this because it shows the artwork and meta data on the TV. The downside is that Plex SUCKS as a music manager. It doesn't even support playlists. I would much rather stream from MM to my TV somehow.

Unfortunately Chromecast does not act as a DLNA receiver, so it doesn't show up in the list of players in MM. Is there any other Chromecast-like device that would work with MM and provide similar functionality as what I have now, including displaying the artwork on the TV?