While I disagree with your orientation, this isn't really that hard as long as you're using a floating floor.

Start by running a line of boards across all three rooms, and orient them so they are as perpendicular as possible to all the walls. (Alternatively snap a chalk line to represent the left edge of that single line of boards.

Now in each room, measure from the left edge (or chalk line) of the board to the left hand wall. Divide that distance by the width of your boards. The remainder is the width of your starter boards in each room. Rip your boards to fit.

Start laying the boards out in the room with the most distance to the left of the line. When you get to your opening, stop - lay the boards out in the next room until they come up to the opening. Run your line of boards from the first room into the 2nd room.

Now since its a floating floor, shift the floor in the new room to align with the board sticking into the room. Click the connecting board into place, and you now have a continuously aligned run.

Repeat for room 3.

While this will not directly answer your question, I'd like to offer an alternative solution.

Modify Existing Molding

If it were me. I'd get some transition molding, with a profile like this.

I'd then set up my table saw to rip the piece, to remove the angle profile on the back edge.

Which would give me a profile like this.

Once I had my molding ripped to size, I'd create a border around the hearth using 45° butt joints.

When installing the flooring, I'd stop short of the hearth to compensate for my expansion gap and the thickness of the moldings leg (the piece that hangs down). Once the flooring was installed, I'd install my molding.

The molding allows the floor to expand and contract, while also providing an attractive transition between wood and brick.

_{Note: If you match the molding color to the flooring color, the transition will be less noticeable.}