The best place to be in June is the North Coast (around Puerto Plata). Punta Cana, on the East Coast, gets in average more rain than on any other month. However, it's still low precipitation compared to most other places in the Dominican Republic.
Get the full story at the following link http://www.hispaniola.com/dominican_republic/info/nature_climate.php
These are all averages taken over many many years. One cannot predict weather very well in the Dominican Republic, not for any part of the country. Take any month of any place in the country and see what the rain was like for that month year after year. The differences are enormous, one year it rains a lot the next year it doesn't rain at all - sunshine every day. In statistical terms, the standard deviation is very high.
However, this is the Dominican Republic, and it is very rare that you will not get plenty of sunshine within a week, or two week vacation. When it rains, it is usually a heavy shower and then the sun shines again.