We are installing recessed lighting in our finished basement. After marking the ceiling (drywall) with the locations for the lights, we cut our first hole and found a wood cross brace mounted diagonally from one joist to the next. It's in the way of our proposed light. We could move the hole, but I am worried that we might run in to this problem with several of our other holes. I am considering just removing the brace. Any thoughts on this?

I am under the impression that the cross braces are installed during home construction to prevent twisting/warping of the floor joists, which helps keep the floor solid and squeak-free. Our home is 30 years old, so I would hope that any warping or twisting is done by now? Do I really need the cross braces anymore?

One more note: These lights are going in below our main living area, where we are about to install new hardwood floors. The existing floor up there squeaks a LOT. We do not know if it is the hardwood, or the sub-floor. We plan to remove the current floor and screw down the subfloor to the joists and hope that takes care of the squeaking. Of course, if we find that the sub-floor is in bad shape, then we'll have to cross that bridge when we get there.

Any advice would be greatly appreciated. Thanks!

The problem is that no one can say for sure what is going to happen once you remove that brace. It could be nothing, or the floor joist could start to "roll", every floor is different. Also, your floor is never done warping or twisting, those braces are as important to that now as they were when they were installed. The most ironic part of this post is that those very braces could be the cause of your squeeky floors.

Cut a little bigger hole in the ceiling, and install a piece of solid bridging, made from the same depth of lumber as your joists, on the side of the lighting fixture nearest the existing row of cross-bridging, while mainatining the clearance from the lighting fixture that the manufacturer's literature requires.

It is likely that they were placed for stability during construction. Once the plywood floor sheathing was fastened to the top of the joists that provides the lateral bracing for stability during the service life of the structure. The sole purpose of the cross bracing at this point in the structures life is to help the floor joists act as a single unit instead of a series of joists. Removing one (or even several) of these cross braces will not be detrimental to the performance of your floor.
I don't know how to add my credentials, but I am a structural engineer.