What is the most cost effective GPU mining rig? I'm looking for specific part numbers for CPU, GPUs, motherboards, cases, etc. 

Assume power doesn't matter. 

Also, what sort of hash rate does it get?
Here are some guidelines to choosing hardware for an effective GPU rig.

CPU: The cheapest you can find. Low-end AMD CPUs are good for this.

RAM: The cheapest you can find.

ODD: Unnecessary, install the OS from a USB flash drive.

HDD: Unnecessary, use a USB flash drive, the cheapest you can find which can hold the OS.

GPU: Basically any ATI series 5XXX+ is suitable. https://en.bitcoin.it/wiki/Mining_hardware_comparison has a list of the hashrate to expect from each. You need to choose one based on the price in which you can get it, the hashrate, and the power (which has implications for electricity costs, and the need for proper PSU and cooling). You need to consider the whole system - using multiple cheap cards can increase the cost of the peripherals.

PSU: Needs to be a good brand and have sufficient power for all the hardware, with some to spare. A 80+ gold PSU is more expensive but could be a good idea, especially if you pay for power. If you're using many graphics cards you can even use multiple PSUs.

Case: If you're using a standard case, it can be cheap but it should have sufficient room and cooling for the cards. Bottom and side fans can help. But the most effective is to use a custom case or no case at all and just custom mounting for the cards.

Motherboard: Needs to be compatible with the CPU and RAM, and depends on the number of cards and the case. If you use a standard case you need PCIe x16 slots with enough spacing between them, and a motherboard with more than 2 suitable slots can be expensive. With a custom case you can use every PCIe slot, including x1, with extender cables - even cheap motherboards have about 5. The available bandwidth for the slots doesn't matter. Other than that there are no special requirements.