The O.P's one is yet another crappy chinese relay board. Optoisolation is fairly pointless if there's a common ground. I'd also give 50% odds there's insufficient creepage/clearance distance to use it for mains, and that the relays can't actually handle 10A. You are pushing your luck if you use it for over 50V AC or over about 15V DC or over 3A!
The one at Gregg's EBAY link looks reasonably nice - the PCB is slotted to increase the creepage distance of the common terminal of each relay. Unfortunately it looks like the ground plane gets too close to the end of the slot so some surgery with an Xacto knife to trim the corners of the ground plane copper at the slots may be required if you want to use it for 230V AC.
It does however over-complicate things with jumpers as all such a relay board needs is two separate connections to the input side of each opto, so you can freely choose whether to drive each channel common anode, or common cathode simply by how you wire the inputs.
Probably the easiest fix for the O.P. without replacing the relay board would be to use a
74HC540 inverting octal buffer between their signal source and the existing relay board inputs, with both its enable inuts tied low.