Here are a few things I've found helpful while constructing t-shirt quilts. As I think of them, I'll update this!
Get it Stabilized
Most clothes used in t-shirt quilts are made from jersey and jersey stretches. In order to neatly sew the shirt pieces to the sashing pieces, you'll need to stabilize the clothes first.
Any non-woven, fusible interfacing will do; however to keep the clothes and seams soft, my stabilizer of choice is Pellon Fusible 80% polyester / 20% nylon non-woven.
Cut the interfacing slightly larger than needed and fuse to the fabric according to manufacturer directions. Cut fused piece as needed and iron again on both sides.
Pick a Sharp
Needle choice is a personal preference. Normally when sewing with knits a ball-point or knit needle is used to avoid holes developing when the seam is stretched. My experience with the stabilized knits is that a quilting or universal needle works the best. Ball-point needles can cause puckers while sewing in woven fabric. Because the knit is sewn to a non-stretch woven fabric (and even when just sewn to another piece of stabilized knit) the seams simply don't stretch enough to cause the holes created by the needle to grow larger than the thread.
Keep it in Order
Even stabilized knits will stretch slightly; to combat this, make sure the woven fabric is on the bottom of the stack while sewing. Most of the stretching is done by the sewing machine's feed dogs. Keeping the woven fabric against the feed dogs and the knit against the presser foot will help the seams retain their shape.
Press it Away
Whenever possible, press the knit fabric toward the woven. This is more of a personal preference, I suppose. Chances are, however, that the knits will have varying weights, so the blocks will look neater if the woven fabric frames the knit.