If you don't strain your body will continue to try and digest the solids (pulp). That's why your cravings are so strong, your system knows there should be more coming down the pike. While you're getting all the nutrition you need, the body hasn't been able to shut down active digestion and get to work on healing itself.
Strain, strain strain. :-) I strain the juice before adding it to they syrup/water so I don't lose any. Sometimes I'll strain twice to be sure.