I don't know why you are assuming i3s are so expensive.
PCPartPicker part list:
http://pcpartpicker.com/p/tQpm23
Price breakdown by merchant: http://pcpartpicker.com/p/tQpm23/by_merchant/
CPU: Intel Core i3-4150 3.5GHz Dual-Core Processor ($104.75 @ OutletPC)
Motherboard: ASRock H97M Anniversary Micro ATX LGA1150 Motherboard ($68.98 @ OutletPC)
Total: $173.73
vs
PCPartPicker part list:
http://pcpartpicker.com/p/8CHMvK
Price breakdown by merchant: http://pcpartpicker.com/p/8CHMvK/by_merchant/
CPU: AMD FX-6300 3.5GHz 6-Core Processor ($94.89 @ OutletPC)
CPU Cooler: Cooler Master Hyper 212 EVO 82.9 CFM Sleeve Bearing CPU Cooler ($29.98 @ OutletPC)
Motherboard: Gigabyte GA-970A-UD3P ATX AM3+ Motherboard ($74.97 @ OutletPC)
Total: $199.84
vs
PCPartPicker part list:
http://pcpartpicker.com/p/HbtdMp
Price breakdown by merchant: http://pcpartpicker.com/p/HbtdMp/by_merchant/
CPU: AMD Athlon X4 860K 3.7GHz Quad-Core Processor ($73.99 @ NCIX US)
CPU Cooler: Cooler Master Hyper 212 EVO 82.9 CFM Sleeve Bearing CPU Cooler ($29.98 @ OutletPC)
Motherboard: Asus A88XM-A Micro ATX FM2+ Motherboard ($52.99 @ SuperBiiz)
Total: $156.96
The Athlon 860k is the clear winner here in value or if money is the biggest concern. The i3 offers the most performance for ~$20 more than the Athlon.
The FX is the oldest, hottest, loudest, most expensive, and flat out the worst in games.