Combine the cream cheese and goat cheese in a bowl. Add the dill and chives.
Spread the cheese and herb mixture onto the pumpernickel slices.
Top the pumpernickel bread slices with cucumbers, strawberries and radishes.
Cut off the crusts of the tea sandwiches and cut the pieces into thirds.
Arrange the tea sandwiches on a platter and chill until ready to serve.
Allow the cream cheese and goat cheese to come to room temperature to make them easier to mix and spread onto the bread.