No, you don't have to go H61. I5 3470 supports 1333 RAM as well. There will be no difference if you used this RAM with either board.
This Intel CPU houses the memory controller, which determines what speed of memory is supported, not the motherboard.
If cost is a concern, the H61 board will work, but like Snowsniper2 said, the motherboard might need to be updated to a newer BIOS version to support your CPU. If this is the case, you would need another compatible CPU to be able to boot the motherboard up so that it can be updated. Then you could install your CPU.
To avoid this potential headache, and the reasons mentioned above, H77 is your best option with the CPU that you have.