Linseman spent his entire junior career with the Kingston Canadians of the Ontario Major Junior Hockey League from 1974–77. He played in 187 games, scoring 286 points (133 goals-153 assists). He also added 33 points (16G-17A) in 25 playoff games. Linseman was drafted by the WHA's Birmingham Bulls in the 1977 WHA Amateur Draft, and spent the 1977–78 with the Bulls, getting 76 points (38G-38A) in 71 games, and adding 4 points (2G-2A) in five playoff games. He was then drafted by the Flyers as their first-round, 7th overall pick in the 1978 NHL Amateur Draft.