F# - 454 (or thereabouts)
Bit late to the game, but can't resist posting my 2d attempt.
Update modified slightly. Now stops correctly if transmitter is hit. Pinched Brian's idea for IndexOfAny (shame that line is so verbose). I haven't actually managed to work out how to get ReadToEnd to return from the Console, so I'm taking that bit on trust...
I'm pleased with this answer, as though it is quite short, it is still fairly readable.
let s=System.Console.In.ReadToEnd() //(Not sure how to get this to work!)
let w=s.IndexOf('\n')+1 //width
let h=(s.Length+1)/w //height
//wodge into a 2d array
let a=Microsoft.FSharp.Collections.Array2D.init h (w-1)(fun y x -> s.[y*w+x])
let p=s.IndexOfAny[|'^';'<';'>';'v'|] //get start pos
let (dx,dy)= //get initial direction
match "^<>v".IndexOf(s.[p]) with
|0->(0,-1)
|1->(-1,0)
|2->(1,0)
|_->(0,1)
let mutable(x,y)=(p%w,p/w) //translate into x,y coords
let rec f(dx,dy)=
x<-x+dx;y<-y+dy //mutate coords on each call
match a.[y,x] with
|' '->f(dx,dy) //keep going same direction
|'/'->f(-dy,-dx) //switch dx/dy and change sign
|'\\'->f(dy,dx) //switch dx/dy and keep sign
|'x'->"true"
|_->"false"
System.Console.Write(f(dx,dy))
