Use a thumbless grip when bench pressing to eliminate shoulder or triceps pain. It also allows you to place the bar a bit lower in your hand – more directly over the forearm bones – and rotate the elbows in more easily.
Using a "false" or thumbless grip when bench pressing is controversial. Many lifters are afraid they'll drop the barbell, which is a very bad thing for your skeletal integrity. Others argue that since you can't squeeze the bar as hard, you'll get a sub-optimal performance because you're not taking advantage of the irradiation effect (contracting a muscle hard leads to a better contraction of the surrounding muscles).
However, through experience I've found that the thumbless grip offers many advantages, one of which is less shoulder strain. I noticed years ago that bench pressing with a thick bar was less stressful on my shoulders. Why? The thick bar forced me to use a thumbless grip.
When you take a regular grip, your hands turn in slightly. This automatically forces you into an internal shoulder rotation position, meaning that the "natural" path you take when lowering the bar will have your elbows pointed outward or flared. This puts stress on the shoulder joint, and if you try to tuck the elbows in – despite the natural inclination for the elbows to be out – you create a lot of torque at the elbow joint. So you either increase the stress on the shoulders or the elbows, neither of which is good.
By using a thumbless grip you can easily keep a more neutral hand position, which makes it much more natural to lower the bar while staying tucked. This reduces shoulder stress without increasing torque at the elbows, resulting in a less stressful bench press.
Try to get comfortable with the thumbless grip if you're prone to shoulder or elbow injuries. Start fairly light for 2-3 workouts to become comfortable with it and give it an honest three weeks to see how your body feels using it.
But, But You'll Die!
Regarding safety, I've used the technique for years for several workouts a week and I have fairly small hands. I use it with thick bars (2-inch), very thick bars (3-inch), and regular bars and have never once came close to losing control. The same is true of all my clients and athletes.
I'm not saying that it can't happen, but honestly, anybody who isn't a total motor moron can become very comfortable with this grip. Any potential accident would likely happen due to bad form caused by using weights that you have no business trying in the first place.