The (3′→2′)-phosphodiester glyceric acid backbone containing an acyclic oligomer tagged with 2,4-disubstituted pyrimidines as alternative recognition elements have been synthesized. Strong cross-pairing of a 2,4-dioxo-5-aminopyrimidine hexamer, rivaling locked nucleic acid (LNA) and peptide nucleic acid (PNA), with complementary adenine-containing DNA and RNA sequences was observed. The corresponding 2,4-diamino- and 2-amino-4-oxo-5-aminopyrimidine-tagged oligomers were synthesized, but difficulties in deprotection, purification, and isolation thwarted further investigations. The acyclic phosphate backbone structure of the protected oligomer seems to be prone to an eliminative degradation owing to the acidic hydrogen at the 2′-position—an arrangement that renders the oligomer vulnerable to the conditions used for the removal of the protecting groups on the heterocyclic recognition element. However, the free oligomers seem to be stable under the conditions investigated.