Based　on　linear　three-dimensional　piezoelasticity,　the　Legendre　orthogonal　polynomial　series　expansion　approach　is　used　for　determining　the　wave　characteristics　in　hollow　cylinders　composed　of　the　functionally　graded　piezoelectric　materials　(FGPM)　with　open　circuit.　The　displacement　and　electric　potential　components,　expanded　in　a　series　of　Legendre　polynomials,　are　introduced　into　the　governing　equations　along　with　position-dependent　material　constants　so　that　the　solution　of　the　wave　equation　is　reduced　to　an　eigenvalue　problem.　Dispersion　curves　for　FGPM　and　the　corresponding　non-piezoelectric　hollow　cylinders　are　calculated　to　show　the　piezoelectric　effect.　The　influence　of　the　ratio　of　radius　to　thickness　is　discussed.　Electric　potential　and　displacement　distributions　are　used　to　show　the　piezoelectric　effect　on　the　flexural　torsional　mode.　The　influence　of　the　polarizing　direction　on　the　piezoelectric　effect　is　illustrated.　For　the　radial　and　axial　polarization,　the　piezoelectric　effect　reacts　mostly　on　the　longitudinal　mode.　For　circumferential　polarization,　the　piezoelectric　effect　reacts　mostly　on　the　torsional　mode.　In　the　FGPM　hollow　cylinder,　piezoelectricity　can　weaken　the　guided　wave　dispersion.