A series of nine hitherto unknown bicylic pyrimidine nucleoside analogues (BCNAs) bearing bicyclic furo[2,3-d]pyrimidin-2(3H)-one, 3H-pyrrolo[2,3-d]pyrimidin-2(7H)-one and 5,6-dihydropyrimido[4,5-c]pyridazin-7(8H)-one bases were prepared in a straightforward approach. The synthesised compounds posses β-D-rybofuranose or β-D-2-deoxyrybofuranose or β-D-arabinofuranose moieties attached to each of the heterocylic ring systems. This is one of a few examples of synthesis of pyrrolo[2,3-d]pyrimidin-2(7H)-one and dihydropyrimido[4,5-c]pyridazin-7(8H)-one nucleosides, and the first example of such nucleosides possessing arabinose moiety. A key synthetic step was a Sonogashira coupling reaction. For coupling with 4-phenyl-1-butyne, we used deprotected 5-iodouridine, 2’-deoxy-5-iodouridine, and 5-iodoarabinouridine and this reaction was followed by cycloisomerization and subsequent conversion of the furane ring into a pyrole ring or a pyridiazine. This approach resulted in the creation of small library of compounds, which were evaluated for their antiproliferative properties against HL-60 and Jurkat E6.1 cell lines. Of all tested compounds, only 3-(β-D-rybofuranosyl)-6-(2-phenylethyl)furo[2,3-d]pyrimidin-2(3H)-one exhibited weak anti-proliferative activity, with IC50 values of 54 and 81 µM for HL-60 and Jurkat E6.1 cells, respectively