Currently you can filter in RadMultiColumnComboBox only by setting the AutoFilter property.

Resolution: You can use the CompositeFilterDescriptors in RadMultiColumnComboBox and filter two or more columns. Please take a look at the code snippet: 
            this.radMultiColumnComboBox1.DisplayMember = "City";
            this.radMultiColumnComboBox1.AutoFilter = true;
            FilterDescriptor descriptor = new FilterDescriptor(this.radMultiColumnComboBox1.DisplayMember, FilterOperator.StartsWith, "Lo"); //first column
            FilterDescriptor descriptor2 = new FilterDescriptor("ContactName", FilterOperator.StartsWith, "Pe"); //second columns 
            CompositeFilterDescriptor compositeFilter = new CompositeFilterDescriptor();            
            compositeFilter.FilterDescriptors.Add(descriptor);
            compositeFilter.FilterDescriptors.Add(descriptor2);
            compositeFilter.LogicalOperator = FilterLogicalOperator.Or; //OR between these columns
            this.radMultiColumnComboBox1.EditorControl.FilterDescriptors.Add(compositeFilter);