Why am I having problems syncing non-itunes music to my iphone?
You need to make sure your iPhone is configured to sync music from iTunes:


Connect your iPhone to your computer
Start iTunes
Select you iPhone in Devices list
Select Music tab
Check the box Sync Music and select what you want to sync to your iPhone (all music / specific playlists)


Also make sure to "Sync only checked songs and videos" option is not checked on your iPhone Summary tab.

On your next sync (automatic or manual, depending on your settings) music should begin to sync to your iPhone.