OK, I was able to put together the following build.

It sits around $800, so there is room for peripherals or more memory or an SSD, etc. etc.

It includes a copy of Windows7/8/8.1 from /r/microsoftsoftwareswap for ~$25. If you don't need Windows, just take that out!

The justification for the parts is below the build. Have fun!

PCPartPicker part list / Price breakdown by merchant

• CPU and cooler: The FX-8320 performs comparably to the often-recommended 4690K but costs $60-100 less, making it a better value choice, especially since games are going to be mostly GPU-bound. The CPU cooler, the Hyper 212 Evo, is one of the best, and best value, cooling solutions for your CPU.
• Motherboard: The Gigabyte 970A UD3P is a great full-featured ATX form factor board with plenty of built in power and cooling for overclocking and getting the most from your CPU
  
• GPU: The R9 290 has about the same performance as the GTX970 but is priced $70-$100 less. I'd like to note that the Anandtech benchmarks were performed with the R9 290's stock cooler, which was Terrible for cooling and noise. Any of the aftermarket coolers (such as the XFX unit in the above build) will get better performance and cooling while running quieter than the original benchmarks.
  
• Case: The Corsair Graphite 230T looks good, has modular HDD bays, great airflow, easy cable management, and comes with 3 fans from factory. If you don't like red fans (or even fans that light up), the BitFenix Spectre Pro (newegg link for pictures - buy from here, just search the model #) is probably the best looking static-pressure-optimized fan that comes in PWM/non-PWM/LED/non-LED models in sizes from 120mm to 230mm. The 230T comes with great fans, but if you need to replace them, those fans will be the best bang for your buck.
  
• HDD/SSD: The hard drive has plenty of space for the price, you can upgrade to an SSD at a later time since it doesn’t improve performance near as much as the other items in the build. Woo!
  
• OS: DO NOT buy Windows7/8/8.1 retail. Get it for $10-$25 from /r/microsoftsoftwareswap from a legit/verified seller. It will be a legit key that comes with your free Windows10 upgrade.
  
• RAM: For most major brands, all RAM is basically created equal. G.Skill is extremely reliable as a brand, and they don't charge an arm and a leg. I chose the single 8GB stick to give you maximum upgrade room and because games don't benefit from dual channel RAM to speak of. If you prefer 2x4GB or want to go with 16GB there's definitely room in the budget.
  
• PSU: EVGA makes great power supplies, and 750W easily powers this build with room for a 2nd GPU or just other additions to the build. It is 80+ Bronze certified so you know it will be efficient and not start a fire in your house. ♥
  

Happy building! :D