Map out your grid pattern on graph paper, using your room's dimensions to determine the width and number of beams. Next, find the center of your ceiling, and snap two intersecting chalk lines, one spanning the length of the room and the other its width. Measuring off those centerlines, snap a series of subsequent lines to match the grid pattern on your graph paper. These lines represent the centers of the beams.
Now start building the U-shaped beams: full-length ones to span the width of the room, and smaller ones to fit in between, forming the grid. Rather than using plain boards and trimming them out later, the design below calls for baseboard molding with an integrated decorative cap for the side walls. The center is a 1x board that's recessed between the sides to create a reveal. For the full-length beams, use tall baseboard moldings. For the small intersecting ones, use shorter moldings so that where the beams meet, their decorative caps will be offset, eliminating the need for complex mitering or coping.
Lift each full-length beam to the ceiling, sleeving it over 2x blocking that's been centered and fastened along your chalk lines. To secure the beams, drive nails through the side walls into the edge of the blocking. Install the small lengths in the same fashion to complete your grid.