Deciding flat foldability of a given mountain-valley pattern is known to be NP-complete. One special case known to be solvable in linear time is when the creases are parallel to each other and perpendicular to two sides of a rectangular piece of paper; this case reduces to a purely one-dimensional folding problem. In this paper, we give linear-time algorithms for flat foldability in two more-general special cases: (1) all creases are parallel to each other and to two sides of a parallelogram of paper, but possibly oblique to the other two sides of the parallelogram; and (2) creases form a regular zigzag whose two directions (zig and zag, again possibly oblique to the two sides of the piece of paper) form nonacute angles to each other. In the latter zigzag case, we in fact prove that every crease pattern can be folded flat, even if each crease is specified as mountain, valley, or unfolded.