Hearts, whose recent financial results showed a debt of £24.7million, had the option to terminate their contracts but Tapping and McGowan have become first-team regulars in recent months.
Australian defender McGowan, 21, has made 21 appearances this season, while former Tottenham trainee Tapping has been a regular feature after making his first start in the Scottish Communities League Cup semi-final win over Inverness in January.
Prychynenko has only made one start for the club and joins Darren Barr and Gordon Smith in being told his services are not required next season.
The 21-year-old told the club's website: "Thanks everyone for all the great memories. I fell in love with Scotland, Edinburgh and Hearts. I will never forget."