Caley Thistle fans face a 150-mile trip to Easter Road to see the clash against Hearts, which kicks off at 12.15pm on February 2.
Inverness were unhappy at playing Hearts at the same venue last season but the repeat journey has been exacerbated by the fact that fans will have limited public transport options at that time for a game which is being shown live on BBC Scotland.
The club said they had attempted to persuade the Scottish Professional Football League to change the kick-off time and location.
Chairman Kenny Cameron said: "It is unfair to ask our supporters to go down the A9 in darkness in February to get to Edinburgh in time for a 12.15 kick-off.
"The directors share the fans' concern and we made verbal and written representations to at least have a later kick-off. But it appears this was not possible.
"We intend having a further discussion with the supporters trust to see if the club can help in any way with the travel arrangements for our fans.
"We suggested alternative venues of McDiarmid Park, Pittodrie or Tannadice rather than making our fans travel to Edinburgh.
"The SPFL have informed us of their decision and, deeply disappointed as we are, we have to accept it."
Davie Balfour, chair of fans' group Caley Jags Together, added: "Following my discussion with the chairman last week we were in unison in our views as to where the semi-final should be played and a trip back to Edinburgh was not on our radar.
"I am alarmed at the prospect of our supporters having to set off in the early morning on a road that is notoriously icy and prone to snow at that time of year.
"Irrespective of distances involved, as we are used to travelling, we feel that safety considerations should be paramount."