The paper Representations of direct products of finite groups.
Burton Fein
Source: Pacific J. Math. Volume 20, Number 1 (1967), 45-58. 

http://projecteuclid.org/DPubS/Repository/1.0/Disseminate?view=body&id=pdf_1&handle=euclid.pjm/1102992967

has what you are looking for and also explains what happens in the case that $K$ is not a splitting field.  Look at Theorem 2.2 and the remark following it.