You shouldn't need more equipment. LX/LH SFPs over single mode fiber should be OK to at least 6.2 miles over good SMF. And inline attenuators are not needed for LX/LH.
Are your patch cables SMF too?
I assume the equipment is already in place. Did you try swapping the SC connectors at patch panel? Because if TX port from one SFP is being connected to TX port of the remote SFP, then you won't get a link at either end.
Also, verify that you're patching into the same fiber strands at each end. (I've seen fiber installs where the two strands at one end on patch panel port #1 are not the same two strands on patch panel port #1 at the other end. One way to test for this is at one site, to "walk" the patch cable SC connectors down each port in the patch panel and watch the SFP for a link LED. Then "walk" them back up with the connectors swapped, in case you had a TX-to-TX situation. You may find the correct strands this way.)
To confirm equipment configuration, can you test both 3560s in the same room, with a single LC-to-LC SMF patch cable? (Or if no LC-to-LC, you can use an SC patch panel and run an LC-to-SC patch into the front of it, and an SC-to-LC patch out the back to the other switch. You may also need to swap SC connectors on one patch cable if you're in a TX-to-TX situation as described previously.) If it works in this "lab" setting then the problem is likely in the fiber between the two sites. If it doesn't work in the "lab" then it may be an equipment or patch cable issue.
