Common words:
* Promise
* Problem
* Possible
* Project
* Perfect
* Present
* Purpose
* Planet
* Picture
* Pattern
* Produce
* Pumpkin
Less common words:
* Prohibit
* Proclaim
* Proceed
* Profess
* Prophetic
* Progress
* Propose
* Protocol
* Pregnant
* Pristine
* Pioneer
* Pirate
* Pacific
Note: This is not an exhaustive list, there are many more 3-syllable words beginning with "p".