No I've never in 48 years heard people say "bursts" of rain. I'm unsure why that particular terminology never was adopted, it's probably more appropriate meaning wise, but Hey! That's English for you! :)

Thanks Lisa. This is the first time I’ve heard the word “cloudburst.” I googled this natural phenomenon and found that a cloudburst is similar to having a huge waterfall pouring down from the middle of the sky. It can even flatten houses and kill people in their sleeps! http://7tab.in/2012/09/cloudburst-in-uttarakhand-20-dead/ Back in Taipei, we have lots raining days. We have thunderstorms but nothing like cloudbursts. The city rains 1/3 of the year so the chance of bumping into a raining day is pretty high. I guess it is nice to live in a place where there are lots of sunny days.

Personally, I wouldn't use either of those in that sentence. If it was certain to rain, I'd write "...with periods of rain..." If it was uncertain but likely to rain, I'd write "... with a chance of rain ..." There is a term called cloudbursts, which are sudden thunderstorms that deliver a lot of rain in just a few minutes and only last a few minutes. I did search for the terms using Google and I see that they are not uncommon, but I have not heard them. I live in Southern California and we don't have very much rain here.