Measure each edge of the floor and mark the middle of each. Stretch a chalk snapline from one mark to the mark across the room from it. Pull up the string and let it snap back down, leaving a chalk line.
Pull the string between the two remaining marks, so it intersects the first line. Set a square at the intersection and adjust the string's position so it's exactly 90 degrees square off the first line. Snap the string as before. You should be left with two intersecting lines that divide the floor into four generally equal quadrants, meeting in the middle.
Set floor tiles without mortar along both lines, starting at the intersection and spanning in all four directions. Put spacers between them. Set all the full tiles that will fit, leaving gaps by the walls where full tiles won't fit.
Measure the spaces left at the walls. If a space between a wall and the last full tile is significantly less than half a tile wide, move the whole line of tiles toward the wall. so you'll have a full tile by the wall and more than half a tile by the opposite wall.
Re-snap your lines to reflect your adjusted layout. When you install the tiles, start at the middle and work your way out to the walls in a grid, cutting the tiles along the walls as needed.