I had the same idea using Sow & Reap, but used Table in place of Do:

First@Last@Reap[
   Table[ 
    If[ j*RandomReal[] > 1/2
     , Null
     , Sow[{i, i^2, i^3}]]
    , {i, 3}, {j, 9}
    ]]

Each time it's executed, the length of output lists varied. For example:

{{1, 1, 1}, {1, 1, 1}, {3, 9, 27}, {3, 9, 27}}

{{1, 1, 1}, {2, 4, 8}, {3, 9, 27}}

{{1, 1, 1}, {1, 1, 1}, {2, 4, 8}, {2, 4, 8}, {2, 4, 8}, {2, 4, 8}}

The iterators i and j could be swapped for a different effect.

That works very nicely. It looks to me like Reap/Sow is a great way when table won't work (as it doesn't for me in this instance). Concise, and functional, thanks!