Tell him to try thanking his guys once and awhile... and mean it. Instead of losing it.
They'll come around.
He just has to ask them to, in a different way.

Sure, no time/no cheque (cause you'd have nothing to go by...why is that by the way?, boss?), but regardless, work has been done. So somehow, payment is due. Can't get around that reality.

Technology may help this 'timesheet problem' along, but it doesn't address the issue, in my opinion... never will.