There is a direct train run by-note this- UZ=Ukrayna Railways, ev 4 days, thus coming off Kiyv via Minsk, TO Riga at rather not-inviting-to-travel hours, back in the mornings. Can even be booked online via uz.gov.us, but only till train departs from Kyiv-the actual tickets are then printed and handed over to chief-provodnik. Ran more often during the now finsihed EURO-Games in Minsk.
BZ=Belorus rails also runs a train Minsk-Riga, but it uses the route via Daugavpils, details not searched yet by me.
There is NO service even with change (as is for Riga-Tallinn) as there remains a gap between last stop in LT and first in LV for local trains.
LUX also had a sub-company named SIMPLE which offered low-budget coachas along same main lines, later these were integrated (as some airlines also do) but now it seems its luxury for anyone! Its coaches mostly start from Berlin-via Warsaw.