Is there an option in SharePoint 2010 where you can set it up so a user can only view the rows they were assigned based on a column in the data list?
You can use the [Me] variable when you configure a filter on a view. It's valid to People type columns.
It will not show the row when the AssignedTo column is targetting a Group. For that you need to go to a more complex solution of using SharePoint designer to configure the filter for the list view.