We consider random linear fields on Z^d generated by ergodic or mixing (in particular case, independent identically distributed (i.i.d.)) random variables. Our main results generalize the classical Strong Law of Large Numbers (SLLN) for multi-indexed sums of i.i.d. random variables. These results are easily obtained using ergodic theory. Also we compare the results for SLLN obtained using ergodic theory and with the help of the Beveridge-Nelson decomposition.