what are your thoughts about going micro atx?
cougar spike micro atx case $35
http://www.newegg.com/Product/Product.aspx?Item=N82E16811553011
amd fx-8320 3.5ghz 8core cpu $160
http://www.newegg.com/Product/Product.aspx?Item=N82E16819113285
gskill sniper 2x4gb 1600mhz ram $55
http://www.newegg.com/Product/Product.aspx?Item=N82E16820231416
asus m5a78-l micro atx motherboard $65
http://www.newegg.com/Product/Product.aspx?Item=N82E16813131942
total $ 315
total without case $280
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alternate parts if you wanted to stay with full sized ATX
asus m5a97 le r2.0 atx motherboard $90
http://www.newegg.com/Product/Product.aspx?Item=N82E16813131872&nm_mc=AFC-C8Junction&cm_mmc=AFC-C8Junction-_-na-_-na-_-na&cm_sp=&AID=10446076&PID=3938566&SID=
any $35-40 case which supports usb3 and atx form factor....
total without case $305
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benchmarks fx-8320 vs i5-3570k (keep in mind this is a $150 cpu vs a $200cpu..)
http://anandtech.com/bench/product/697?vs=701
benchmarks fx-8320 vs i3-3220 (this is a more realistic comparison $150 cpu vs $150 cpu)
if you dont overclock this requires about 280w for psu (so that 420w should work)
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if you wanted to spend more towards your $350 limit.... it might definitely be worth going with an intel build instead
cougar spike micro atx case $35
http://www.newegg.com/Product/Product.aspx?Item=N82E16811553011
intel i5-3570 3.4ghz 4-core cpu $210
http://www.newegg.com/Product/Product.aspx?Item=N82E16819115233
gskill sniper 2x4gb 1600mhz ram $55
http://www.newegg.com/Product/Product.aspx?Item=N82E16820231416
asus h61m-a usb3 micro atx motherboard $65
http://www.newegg.com/Product/Product.aspx?Item=N82E16813131963
total $365
total without case $330
as for performance see the first benchmark.
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honestly you could go either route. both of the builds i listed are better than the i3.
the amd build is cheaper for less performance (but more performance than similar cost intel builds) while the intel build is going to be more powerful than the fx8350. the h61 chipset on the motherboard means that overclocking isnt possible so i gave you a normal locked i5 chip (not a k version). if you were into overclocking for less price you could get more performance out of the amd system. you would want to get a better cpu cooler though for this.
with the cash you saved by going amd (okay.. probably need just a little bit more than this...) you could add in a boot ssd drive to increase your system speed by quite a bit. this would give you a much snappier computer. personally though instead of just a "boot" drive (where you cannot store anything) i would get something of a half decent size (128gb) so that you could put a few programs on it as well.
edit: reason i didnt say about overclocking i5-3570k (if you went with a k version for $10 more) is that to get an overclockable motherboard you would be looking at more cash as well so its not terribly budget friendly to go intel AND overclockable