Can I do anything about a grenade in Mass Effect 3?

Every time I see a grenade fall near me, I try to pick it up and throw it back. This leads to a badly overcooked Shepard pie.

Is there anything I can do about a nearby grenade other than run away? Can I deactivate it, throw it back, or push it away somehow?

I have yet to see anything other than move away as an effective grenade counter.

This opinion is borne out by several more days of game play, the recipe for Not-Shepard-Pie is move it, move it, move it!!

Running away is the best option. However if you're fast enough on the trigger and you see an enemy priming to throw a grenade, a quick headshot or burst of fire to kill him before he completes his throw animation will result in the grenade going straight up in the air and coming back down or falling short of your location. If it lands in front of your cover then don't move. Your cover will protect you.