I thought of this idea, and can't seem to find it anywhere here. How about using Thread Stands for cookie cutters. It won't work if you have gobs of cutters. You need more space. Or, you could use the racks for drying your cutters after washing. Here's a couple of examples.
You can find this one at connectingthreads.com or I've seen this type at Wal-Mart
There are many other versions online. Go to Google Images and put Thread Racks in search. Just an idea.