Do your floors have baseboards? Cabling is frequently hidden behind them. It might be a bit involved at corners and door jambs, but if I'd have to put down a cable without going under floor/inside walls, my first bet would be trying to put it behind baseboard.
Firstly, it looks much better (or rather, it doesn't, as the cable is all hidden!).
Secondly, you don't need anything much for it, except a screwdriver (power one for preference). Unscrew the screws that hold baseboard to wall, lay down the cable and screw the board back. Repeat for next section. Flat cable is excellent for this.
This does assume that baseboards are screwed to the wall rather than nailed or glued. In my limited experience, this is usually the case. The only issue might be that screws could be painted over and you'd surely scratch paintv of them while getting them off/on.
At door jambs it can be problematic, but there are various options depending on actual jambs. In my last place, I could just push the cable under the bottom of the jamb.
You should, of course, make sure that the whole pathway is useable before attempting this. It'd suck if you were halfway with the cable, and suddenly saw that the next section is impossible to put under board.
Also, if you are going this route, perhaps you can significantly cut your cabling work and length by actually doing one or two holes between rooms -- you could have the holes behind baseboards too.
Alternatively, if you don't have baseboards, you can put cable trunking (not sure if there US term is the same but I am referring to the "u shaped panel thingies" you mentioned) instead. Basically self adhesive plastic sheaths for cabling that you stick to the walls. It does look better if you are leaving ceiling alone in any case, and route your cable art floor level.