Syncing music without iTunes...or can you?
I would try out Google Music. It will be the most native with your device, and gets relatively close to syncing, in theory.

The first thing you will have to do is go to the Google Music Homepage and download the Music Manager tool. Once downloaded, you tell the Music Manager where your music is on your computer, and it uploads it all to the cloud!

This music becomes visible as soon as you log into Play Music on your phone with the Gmail account that you used to set up Google music on your computer.

Then whatever music is placed on your computer in your music folders is accessible at music.google.com and on all of your Android devices.

The difference is, your music does not actually sync to your phone, but is streamed when you choose it on your mobile device.

This can be slightly modified by selecting an album or such on your device, clicking the little expansion arrow (usually at the bottom right of the item), and choosing the Offline music option (it has a picture of a pin next to it). Your phone will then download this music so it does not have to be streamed to play later. 

You would have to do this with all of the music you do not wish to stream, but if you have a decent internet connection, streaming is very much a viable option. And saves TONS of space on your phone!

Hope this helps!