The previous posts offer the best solutions, but many stitchers also use a laying tool to lay their threads. After you come up through the fabric you stroke the thread until it is untangled and then put the needle down while still stroking. You may see this technique on Shay Pendray's needlework studio that is aired on Public Television. It works wonders when using multiple plys of thread. Amy Bunger's needlework videos may also have it demonstrated on them.
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twisting thread trouble by: Valerie Kalyani
There are two ways to deal with this. The most common one is to let your thread and needle hang freely every few stitches and untwist itself. The other (which I prefer) takes a bit of practice. As you sew, when you take hold of your needle to pull to the front of your work, twist it a teeny bit, about a quarter-turn in the opposite direction it wants to twist. You will have to practice a while but if you master this simple technique you will not have twisting thread any more and will only rarely have have to dangle it to untwist.