Rohit's solution gives a Root object, which may not be desirable.
To do the selection after the fact, use
In[1]:= soln = Solve[a t^3 == b, t]
Out[1]= {{t -> b^(1/3)/a^(
1/3)}, {t -> -(((-1)^(1/3) b^(1/3))/a^(1/3))}, {t -> ((-1)^(2/3)
b^(1/3))/a^(1/3)}}
In[2]:= Select[soln,
Simplify[Element[(t /. #), Reals], Assumptions -> {a > 0, b > 0}] &]
Out[2]= {{t -> b^(1/3)/a^(1/3)}}
