I'll have to say up front that I'm no good at all at tiling problems. I did manage to find a solution for this by myself, so I'll include the thought processes that I used to get there, for what they're worth.
My first thought is how the bottom of the grid is going to be tiled by these four pieces. My assumption was that all four pieces were going to have to reach down toward the bottom of the grid. Based on that, my second assumption was that there would be a vertical line going down the center of the grid in the final solution. [1]
With these two assumptions in place, I'm looking for a radially symmetric way to split the following grid in half so that A and A' are in two different parts, as are B and B'.
A is in the same piece as either B or B'. Any piece that includes A and B' and has nine pieces will separate A' from B, though. So A connects to B and A' to B'. Putting these together and drawing the radially symmetric lines leads us quickly to a solution:
In the actual puzzle grid, it looks like this:
The shaded squares belong to ABBDDC.
Notes: I didn't even look at this during the test. I almost appealed to the general community to solve this for me, I'm so bad at these problems (and if someone wants to post a rigorous solution, I'll be happy to take it).
[1] Nick Baxter tells me that these are very unusual and fortuitous assumptions to make, since the solution to most tiling puzzles are often rotationally symmetric, and asked me if I could either justify my assumptions further or make it clear that I'm really lucky. Well, I'm really lucky. Clear enough?
Seriously, my first assumption was that the four pieces were all going to be 90 degree rotations of each other, but then trying to find a nine-square piece that included C, a corner square, and a center square made me look elsewhere. Then I went to the diametrically opposite viewpoint; that maybe the four pieces would have no symmetry at all, but then the problem would be completely out of my league. So I made the assumptions I did not because I thought that they would lead to the solution so much as because if they didn't lead to the solution, then I wouldn't be able to solve the problem.