Peptoid molecules are biomimetic oligomers that can fold into unique three-dimensional structures. As part of an effort to advance computational design of folded oligomers, we present blind-structure predictions for three peptoid sequences using a combination of Replica Exchange Molecular Dynamics (REMD) simulation and Quantum Mechanical refinement. We correctly predicted the structure of a N-aryl peptoid trimer to within 0.2 Å rmsd-backbone and a cyclic peptoid nonamer to an accuracy of 1.0 Å rmsd-backbone. X-ray crystallographic structures are presented for a linear N-alkyl peptoid trimer and for the cyclic peptoid nonamer. The peptoid macrocycle structure features a combination of cis and trans backbone amides, significant nonplanarity of the amide bonds, and a unique "basket" arrangement of (S)-N(1-phenylethyl) side chains encompassing a bound ethanol molecule. REMD simulations of the peptoid trimers reveal that well folded peptoids can exhibit funnel-like conformational free energy landscapes similar to those for ordered polypeptides. These results indicate that physical modeling can successfully perform de novo structure prediction for small peptoid molecules.