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In general, I kinda of shirk when I hear "budget" and "gaming" computer in the same sentence. It's like trying to build a budget race car. But you can still get decent enough performance without spending a ton of cash.
My personal recommendation for anyone considering gaming on Intel is to go with the P67 chipset, since it allows you to overclock should you choose to do so, in addition to SLI/Crossfire and other high performance options.
A lot of people I've been talking to have been looking at budget Intel builds with Core i3 CPU's, these are price-comparable to Phenom II's (although you only get half as many cores), and H61 motherboards are price comparable to Socket AM3 boards.
I admit, I haven't really ever considered the H61 as a gaming option. I always rather shoehorned it as the low-end office/productivity chipset option for people who wanted a light computer and were probably just going to be using Intel On-Die video anyway.
So, the technical difference between the H61 and the H67 only appears to be a couple of SATA ports and a couple of USB ports (most of which go unused anyway), and the software-based Intel RAID support.
So, if you can't afford to go P67 and your dead set on Intel, is there any real reason to go H67 over H61 (aside from motherboard-specific differences)?