Here is some code that will give you what you want.
#Example Data
n = 6
time02H = abs(rnorm(n))
time06H = abs(rnorm(n))
time24H = abs(rnorm(n))
allPAL.DE = data.frame(time02H,time06H,time24H)
plot(0,0,xlab="Time",ylab="Values",type="n",xlim=c(0,2),ylim=c(min(allPAL.DE,0),max(allPAL.DE)),axes="FALSE")
box()
axis(1,0:2,c("time02H","time06H","time24H"))
axis(2)
abline(h=0,col="grey",lty=2,lwd=2)
for(i in 1:nrow(allPAL.DE)){
lines(0:2,allPAL.DE[i,],col=i,lwd=2)
}
You could add different options, colors and text to make the plot look nicer but its the basic idea that you were asking for.