The rainy season starts in late April to early May normally. There is water in the Chiriqui Viejo watershed all through the dry season. You can easily do 4 days of class 3/4 water on that river and do different sections each day. It has good flows (probably about 800-1000 cfs on the lower section right now). If you repeat some of the sections, then it would be fun to spend a week on that river. In the rainy season, the river is mostly class 4 to 4+.
Starting in May, most of the bigger rivers will be runnable. If you want the steeper creeks, September-November is the best, but there is good paddling June-August also.
December can be really variable. Some years there is lots of water, some years only the biggest drainages have water. From January through April, you mostly have the Viejo and some class 2 runs.
Of course, I am speaking about the Chiriqui province which is in western Panama (borders Costa Rica and the Pacific ocean). The best paddling is here, but there are good runs in other parts of the country as well.
Feel free to email me if you have more questions.