Hi again Carrie - you're certainly busy at the mo!
Anyway, raspberry canes. Are yours summer or autumn fruiting varieties? I ask because there's a bit of a difference in the way you treat them. Mine are all autumn fruiters, so what I did was plant them out (last month), then last week I cut them down to the ground and chucked a load of rotted manure over them as a mulch. This is what the old boys at my allotments advised, so I reckon it must work. I don't think the weed supressing membrane is a good idea for the reason you explained.
Dunno much about redcurrants (just planted mine as well), but I think they are hungry feeders so competition from other plants probably a bad idea. Chuck manure at em as well I reckon. That's what I'm going to do.