I prepared some short 1 inch thick wood "sandwiches" using four 1/4 inch cedar boards each, glued together with polyurethane construction adhesive. They are now fully cured (>1 month) and I am ready to cut them into short pieces as spacers. I plan to use a standard/small circular saw with relatively low tooth count carbide framing blade. Do I have to worry about anything like kickback or gumming up when the blade goes through the cured adhesive layers?
Having never worked with urethane glues, I can't say for certain, but I'd have a hard time believing that this will be any more difficult than cutting solid lumber, plywood, MDF or anything else. If anything, I'd imagine that you might get some residue build up on the blade, the same as if you were cutting a resinous wood, like fresh pine with lots of sap.
If you do get buildup, your cut speed & quality will go down as the resin builds up on the cutting edge of the teeth, effectively dulling them. If that happens, there are quite a few techniques for cleaning the blade, from general purpose cleaners to products specifically formulated for cleaning saw blades. I'd suggest a perusal of Woodworking.SE to find pointers on cleaning your blade, should it be necessary.
As a reference point, I've used an 80-tooth blade in my miter saw for cutting vinyl siding for my house. It's not cutting well at the moment because the blade is an absolute mess from the melted siding sticking to it, however it does still cut (wood, PVC pipe, etc.) I just have to go slowly and be patient. I do really need to clean it, but haven't taken the time to do so yet.
As a side note, a low-tooth-count blade will not give you the cleanest cut, so either that's not necessary in this situation or be prepared to do some clean up work.