A new type of indefinitely growing self-avoiding walk is introduced in which the walker is not allowed to perform a step pointing towards a site he has already visited. This restriction leads to a walk which is a succession of long directed parts. A Monte Carlo study on the square lattice suggests that the radius of gyration exponent nu =1 in two dimensions. The enhancement factor exponent gamma =1 in all dimensions. A self-consistent method gives nu =1 when 1<or=d<or=2, nu =2/d when 2<or=d<or=4 and nu =1/2 above dc=4, the upper critical dimension.