An irreducible open 3-manifold W is R²-irreducible if it contains no non-trivial planes, i.e. given any proper embedded plane in W some component of W −II must have closure an embedded halfspace R² ×(0, ∞). In this paper it is shown that if M is a connected, P²-irreducible, open 3-manifold such that π₁(M) is finitely generated and the universal covering space M of M is R²-irreducible, then either M̃ is homeomorphic to R³ or π₁(M) is a free product of infinite cyclic groups and fundamental groups of closed, connected surfaces other than S² or P². Given any finitely generated group G of this form, uncountably many P²-irreducible, open 3-manifolds M are constructed with π₁(M) ≅ G such that the universal covering space M is R²-irreducible and not homeomorphic to R³; the M are pairwise non-homeomorphic. Relations are established between these results and the conjecture that the universal covering space of any irreducible, orientable, closed 3-manifold with infinite fundamental group must be homeomorphic to R³.