TL;DR: Unless you're specifically looking for maximum bulk transfer speeds, you don't need the 6Gbps link and most SSDs aren't that fast anyhow. You'll still get a good speed boost even with 3Gbps SATA links if you go from HDD to SSD.
The SATA controller on the mobo definitely impacts the maximum speed of your SATA ports. If you can't find SATA speed or version (SATA2 is 3Gbps, SATA3 is 6Gbps) specs for your mobo and it's more than a few years old, there's a good chance that it doesn't support 6Gbps.
Mind you, a large number (the majority?) of consumer SSDs can't actually fully take advantage of 6Gbps (750MB/s) either. They tend to top out at around 550MB/s (4.4Gbps). There exist some very high-end SSDs that will push the 6Gbps boundary, but you can still get much faster storage by going with a cheaper SSD and using it with SATA2.
Most of the speed advantage of an SSD (for typical PC use, as opposed to bulk data anything) is the instant seek, not the bulk I/O speed. Even a magnetic HDD can hit 200MB (1.6Gbps), which is almost certainly loading data faster than you can actually process it but is probably barely more than half of what your SATA ports could handle. That's sequential I/O, though; throw in seeking, and the speed of a HDD drops drastically.
Thus, if you got a SSD, you'd still have a huge boost in speed. 3Gbps is almost twice as fast as that HDD mentioned above (and 6Gbps nearly 3.75x), but that's not the important factor. The main speed advantage of an SSD is that, no matter where the data is on disk or even really how fragmented it gets, the speed doesn't go down with an SSD. For a HDD with a 10ms average seek time, loading 100 1-byte files takes longer than loading a single 100 MB file, even though it's a million times as much data. An SSD (even at 3Gbps) would load the 100 MB faster than the HDD, but it would load the hundred tiny files much faster.
