I think the first step is usually to iron a fusible interfacing to the back of the t-shirt fabric to stabilize it. My choice for this would probably be fusible tricot because it is lighter in weight than most fusibles (both woven and unwoven) and allows the fabric to retain some drape. Many fusible interfacings make the t-shirt fabric unpleasantly stiff.
I have purchased fusible tricot at JoAnn's, but have found it online from a drapery supply company for considerably less (and the fusible is lighter weight too, which I like). Try Googling "fusible tricot" and drapery to find some sources.
Anyway, you would cut the seams from the front or back of the shirt and then fuse the flat piece of fabric from the front or back to the fusible tricot before cutting it. Usually you cut the t-shirt fabric into a large square or rectangle to best display the logo.
Most people use regular cotton fabric for sashing between the t-shirt blocks.