  Let $n$ be an even natural number. We compute the periods of any
$\frac{n}{2}$-dimensional complete intersection algebraic cycle inside an
$n$-dimensional non-degenerated intersection of a projective simplicial toric
variety. Using this information we determine the cycle class of such algebraic
cycles. As part of the proof we develop a toric generalization of a classical
theorem of Macaulay about complete intersection Artin Gorenstein rings, and we
generalize an algebraic cup formula for residue forms due to Carlson and
Griffiths to the toric setting.
