Archaic kantele tunes, playable on 5 strings:
Konevitsan kirkonkellot (you can listen to it here, played by Helena Tuupanen)
Mistä koppa kantelohon
Satavuotinen sakka
Lehmän tanssi

Archaic jouhikko tune, playable on the bowed lyre (jouhikko):
Grannens bastu (The neighbour's sauna), with drone on the "a". (You will find some versions on youtube, played on jouhikko and guitar).

Medieval tunes from Piae Cantiones, a collection of songs which were sung in the cathedral school of Turku from 13th to 16th century (published in 1582):
Scribere proposui
In stadio laboris

Folk tunes (pelimannimusikki) which work fine on harp or kantele:
Vanha häämarssi pohjanmaalta
polska från Alajärvi efter Johan Gustaf Hauta-Aho
Pinnin Valssi
Lampaan bolska (The sheep's polska)

Luurankojenkka (Skeleton dance) as it was sung by Yrjänä Ermala in Turku 2009. It has interesting, not quite romantic lyrics...