It does run but be warned that till July Latvian Rail are running buses for part of the route from Riga to Valga and this means you miss the connecting train (journey time Riga to Valga around 4hr). The bus arrives Valga at 1646 and the last Tallinn train leaves at 1642!! The only option now is Riga 0620 arrive Valga at 1026. Leave Valga at 1400 arriving Tallinn at 1716.
In the other direction it's the same problem with the main morning connection lost so there's only an afternoon one now Tallinn 1309 ar Valga 1630. Depart by 1703 on replacement bus to Valmeira then train arrives Riga at 2050
Try
http://www.pv.lv/lv/kopejais_iekszemes_vilcienu_kustibas_saraksts/
and select "Kopējais iekzemes vilcienu kustības saraksts" to get a complete Latvian Rail timetable till July for all routes in Excel format.
Also be aware Elron have imposed 80kph speed limits from Tartu to I believe Jogeva which is slowing them down too.
Riga to Valga is 5.60euro one way and Valga to Tartu is 4.40 euro and Tartu to Tallinn is 10.90 euro with First Class only a little more.
LuxExpress buses are 10-25 euro and Simple Express start at 3euro and buses only take 4.5 hours.