Moving target path finding

In working on implementing better pathfinding than “which way is it? ok go that way.” I decided to start with A* as I’ve heard it mentioned before. A* might be a bit overkill for a grid-based game where you can only move in four directions, but oh well. Unfortunately after getting it working I discovered that a basic A* implementation doesn’t work very efficiently when the target is moving around. So, I’m trying to figure out a better way to get my enemies to find their way to the player with out getting stuck. If anyone has suggestions, or experience with doing something like this I’d love to read it ;D