Angle a wood flooring strip into the space between the last installed strip and the heat register's edge. Line up the strip's front edge with the heat register and allow its back edge to overlap the previous row of strips. Slide the strip down the floor so it is pressed tight against either the wall or the last full strip you installed.
Make a light pencil mark on the strip to indicate the heat register's left and right edges. Remove the flooring strip. Measure the space from the previous row to the register's front edge. Transfer this measurement to the flooring strip you just marked, measuring back from the marked edges.
Draw a line along the wood flooring strip, from the left mark to the right mark, to indicate the portion of wood you will remove using a straight edge. You will end up with a small rectangle marked on the flooring strip's edge. Use a jigsaw to cut the flooring strip along the lines you drew.
Position the wood flooring strip between the previous row and the heat register, with the register fitting snug in the rectangle you cut out.
Cut another wood strip short enough to fit between the space between the heat register's left or right edge and the wall or previously installed strip.
Repeat Steps 1 through 3 to cut a rectangle out of a flooring strip to be installed along the heat register's other side.