They are tedious. They take about a second each. In the later missions, that can mean 5-9s at the end of every alien turn because of all the dread. It gets old and slows down the game for no good reason.
My suggestion: effects log, left side of screen:
Barry Allen is overcome with dread.
Bruce Wayne is unnerved.
Clark Kent is overcome with dread.
Oliver Queen has gone berserk!
And so on. Clear the log at the end of the human turn. That would be more useful than a message that is gone a second later. Ideally those messages would be hyperlinkish -- they'd set focus on the affected soldier when clicked.