The main focus in this paper is the algebraic K-theory and higher Chow groups of linear varieties and schemes. We provide Kunneth spectral sequences for the higher algebraic K-theory of linear schemes flat over a base scheme and for the motivic cohomology of linear varieties defined over a field. The latter provides a Kunneth formula for the usual Chow groups of linear varieties originally obtained by different means by Totaro. We also obtain a general condition under which the higher cycle maps of Bloch from mod-lv higher Chow groups to mod-lv étale cohomology are isomorphisms for projective nonsingular varieties defined over an algebraically closed field of arbitrary characteristic p [ges ] 0 with l ≠ p. It is observed that the Kunneth formula for the Chow groups implies this condition for linear varieties and we compute the mod-lv motivic cohomology and mod-lv algebraic K-theory of projective nonsingular linear varieties to be free ℤ/lv-modules.