I would take the ASRock 970M Pro3 (currently at $50 AR at Newegg) with a FX8320E @ 4-4.4GHz over any H81 + Core i3 Haswell.
ASRock 970M Pro3
6x SATA 6Gb/s with RAID support
4x Memory slots DDR-3 2400MHz up to 64GB
Support up to 140W TDP CPUs
2x PCIe 16x slots Gen 2.0 with CrossFire support
USB3 Front Header slot
vs
H81
2x SATA 6Gb/s
2x Memory slots DDR-3 1600MHz up to 16GB
1x PCIe 16x Gen 2.0 slot
USB3 Front Header slot (only from $50 price and up models)
With latest games supporting more than 4 threads and DX-12 games on the horizon, going for the Core i3 now is a no no unless you really want to play a game like Dayz only.
i3 supports 4 threads, and OP says he will not likely overclock. Yes, I might give a nod to the 8320E if OP planned to invest a little more in a nicer motherboard, aftermarket cooler, slightly bigger power supply, and crank it up to 4.4, but stock vs stock? It's a tossup.
@OP, you'll find benches supporting both sides. The reason it's a hard decision is that they're they're roughly equal but different.
~
I went onto gameGPU and opened up each genre in a tab, and picked the top / most recent review for each section. Here are their CPU charts:
In 4/5 games, the i3 beats the FX-6300, though it's very close in one test.
In 1 game the i3 beats the 8150, roughly ties in 3/5 games, and loses in one game.
In 3/5 games, the FX-8150 beats the FX-6300, though they're often close.
In 4/5 games, the i3 loses to an FX-8350.
^ This is a small sample, but they're all recent releases. What this suggests to me is that these games are mostly well threaded
already, since the older, slower 8150 is beating or matching the FX-6300. Even so, the i3 is approximately in between an FX-6300 and an FX-8350. I would expect it to be a tossup when compared with an FX-8320E, which comes out below an 8150 in some tests, and above it in others.
So again, no wrong choice here.