AM3+ CPU requires AM3+ motherboard.
AM3 CPU can use AM2/AM2+/AM3/AM3+ motherboard.
AM2+ CPU requires AM2+ motherboard.
AM2 CPU can use AM2/AM2+ motherboard.
The reasons are pretty simple. AM3+ sockets are made for the higher loads of the FX series chips such as the 8 core 8000 series. AM3 sockets are up to a six-core only and thus there's a big difference in the loads. If you were talking about a normal FX 4/6 core (FX-4300/6300) it would theoretically possible if the motherboard manufacturer updated the BIOS to support it since the thermals and cores are similar to the AM3 Phenom 4/6 core CPUS. That's a big if though. It looks like Gigabyte did make that motherboard in an AM3+ style and had support for the FX-Bulldozer chips - but not the Piledriver series.
Ultimately though, if you want to keep your AM3 board, you'd be best served by getting a high end 6 core Phenom II chip. Your board supports the Phenom II x6 1100T which is a pretty damn potent CPU. With a good cooler it can OC pretty well and returns a geekbench score of mid 11,000 which puts it in i5-4670 territory. Yes, it uses more power, but they're also drop-in replacements for what you have, and a quick eBay search turns them up in the $150 range.