I am planning on laying down a tile floor in a pattern. The patter looks like this:
http://www.bedrosians.com/flrpatt9.htm
the only difference is my tiles will be smaller - 6x6, 6x12, 12x12, 12x18
the floor we will be tiling is 150 square feet. (although the room is not square).
I do not know how to figure out how many of each tile I will need to order since the pattern is not squared off.

Any suggestions?

By my calculation, using the size tiles you mentioned, one unit would be a 3x3' square, or 9 square feet. The amount of each size tile should be the same in that diagram for the tile size you are using. For 150 sq ft you would need enough tile to make 17 units (9'x17units= 153 sq ft). So count up the number of each size tile making up one unit and multiply by 17 to get the amount you need. Get a few extra to allow for cuts along the walls. And you can cut the bigger ones down to a smaller size tile if you run short.