After a bit more research:

It was a favorite song of Masaryk--that seems to be the only direct tie to him.

Here's a Moravian version:


Teče voda, teče

1. Teče voda, teče, přes velecký majír,
něhal si ma něhal, starodávný frajír,
něhal si ma něhal, starodávný frajír.
[The second line of each verse is similarly repeated.]

2. Něhal som ťa, něhal, dobre ty víš komu,
co ty reči nosí, do našeho domu.

3. Do našeho domu, pod naše okénko,
co sem se naplakal, sivá holuběnko.

4. Vráť se, milý, vráť se, od kysúcké vody,
odnesl's mně klíček, od mojí svobody.


Two additional verses from a Slovak version:

5. Skúr sa Stará Turá, v kolečko obrátí,
sloboděnka moja, ta sa nenavrátí.

6. Už sa Stará Turá, v kolečko obracá,
sloboděnka moja, ta sa nenavracá.


The song seems to stem from the Horňácko region of south-eastern Moravia, near the border of Slovakia. Three place names are indicated:
(1) The Velička River, a tributary of the Morava southeast of Brno (and of the town of Stražnice).
(2) Stará Turá, a town in Slovakia, and (judging from context) a nearby peak in the White Carpathian mountains which separate Moravia and Slovakia.
(3) The Kysuca River, in the Kysuce region of northwest Slovakia, north of the town of Žilina.

The song text is vague in context, and the language is in dated dialect, so that even Czech people who have commented on the song are unsure how to interpret it. I'll give my faulty translation, and then describe some of the conjectures.

1. The water flows through the farm;
you left me, my former beau.

2. You left me, you know well to whom
[that you courted] at our home.

3. At our home, under our window,
why did you cry [so much?], dove-grey dove?

4. Return, my love, from the Kysuca water;
you took away the key to my freedom.

5. [Even] when Stará Turá turns around in a circle,
my freedom, you will not return.

6. Now Stará Turá turns around in a circle;
my freedom, you aren't returning.


Notes:

majír = farm or manor garden; derived from German "Meierhof" (manor farm)

frajír = milinec/boyfriend; starodávný literally means ancient, but here it seems to mean "of old" or one-time/former, or possibly disheveled.

reči nosí: much confusion here, not only about the meaning but about who's doing what or talking of whom. Literally, courtship/news/speech/rumor carries/brings.


The young woman had a boyfriend who courted her for some time, but her family didn't approve or denounced/talked him down because she was "to the manor born" while he was common. So he left and went to Kysuce. Possibly he abandoned her for another woman, but the mention of the Kysuca waters, combined with his never returning may mean that he drowned himself in that river.

Taking the key to her freedom may mean that he left her "spoiled goods". Or it may simply be that her family has meanwhile arranged a "suitable" but loveless marriage for her. Or that she remains pledged to him till he returns--if ever. I take the final verse, with the mountain turning, to mean she is about to take her own life.

One translation I found puts a different spin on the story: he has left to fight for freedom, and she keeps vigil until he returns. Finally, the mountain turns, he's victorious and she awaits his return and her freedom. I haven't encountered a Czech or Slovak text which supports this reading, but it sounds like a likely Soviet-period recasting.

The water flowing at the beginning of the song can also be taken as a metaphor for her tears.