Many years ago when I was a teacher, a fellow teacher shared with me that a used fabric softener sheet does wonders at getting the marks off of a dry erase board. The next time you do laundry, simply save the used fabric softener sheet.
When you need the marker from your dry erase board erased, use the fabric softener sheet (instead of an eraser) and your done. If you have some smudges and marks that have been on your board for a while, using this method will require some elbow grease.
I really prefer to use the fabric softener sheet on a daily basis for regular erasing. If you have some long term smudges, you are hoping to get rid of, you really need to try this next method.
Rubbing Alcohol:
All you need here is a little rubbing alcohol (the 70% alcohol you have in your medicine cabinet will work perfectly) and a soft microfiber cloth. I like to buy the lint free cloths from Dollar Tree or you can buy similar ones from Amazon here.
Simply dampen the end of your soft cloth with rubbing alcohol and wipe the smudges away. I was so amazed with how well this worked!