The killing is done in several steps.
First, the convicted are given a high dose of barbiturates (about 10-15 times the normal dose), which causes them to fall unconscious within a few seconds.
Then pancuronium, and finally a massive overdose of potassium chloride are injected. The former is similar to curare (indio arrow poison) in that it paralyzes muscles (thus causing apnea). If the person was not already unconscious at that time, this would be very inconvenient (not strictly painful, but nevertheless really really uncomfortable).
The latter (potassium chloride) is actually harmless, but in this massive overdose causes a cardiac arrest and convulsions. Which, again, would be very painful, if the person was not already unconscious, and under pancuronium.
So, in short, yes. Painless.