Pg. 234 PHB, highlights are mine..
You whisper a discordant melody that only one creature of your choice within range can hear, wracking it with terrible pain. The target must make a Wisdom saving throw. On a failed save, it takes 3d6 psychic damage and must immediately use its reaction, if available, to move as far as its speed allows away from you. The creature doesn't move into obviously dangerous ground, such as a fire or a pit. On a successful save, the target takes half as much damage and doesn't have to move away. A deafened creature automatically succeeds on the save.
Assume my character is grappled and has successfully cast Dissonant Whispers. Also assume the creature failed its saving throw and has a reaction available.
Does the creature release the grapple in this scenario?
I found this text on pg. 234 PHB which explains the rules for the Grappled condition:
... The [grappled] condition also ends if an effect removes the grappled creature from the reach of the grappler or grappling effect, such as when a creature is hurled away by the thunderwave spell.
Also, on pg. 195:
Moving a grappled creature. When you move, you can carry or drag the grappled creature with you, but your speed is halved...